Adam Mickiewicz University, PoznańFaculty of Mathematics and Computer Science

Uniwersytet im. Adama Mickiewicza w PoznaniuWydział Matematyki i Informatyki

PhD dissertationMathematical sciences

Mathematics

Rozprawa doktorskaNauki matematyczne

Matematyka

Structure constants of Jack charactersStałe strukturalne charakterów Jacka

Adam Burchardt

SupervisorPromotor

Prof. dr hab. Piotr Śniady

Auxiliary supervisorPromotor pomocniczyDr Maciej Dołęga

Poznań 2018

Research supported byNarodowe Centrum Nauki

Grant number2014/15/B/ST1/00064

Moim rodzicom

Abstract

In 1996 Goulden and Jackson introduced a family of coefficients (cλµ,ν) in-dexed by triples of partitions which arise in the power sum expansion ofsome Cauchy sum for Jack symmetric functions J (α)

π . Goulden and Jack-son suggested that there is a combinatorics of matchings hidden behind thecoefficients cλµ,ν . This Matchings-Jack Conjecture remains open.

Jack characters are a generalization of the characters of the symmetricgroups, they provide a kind of dual information about the Jack polynomials.We investigate the structure constants gλµ,ν for Jack characters. They are ageneralization of the connection coefficients for the symmetric groups. Wegive formulas for the top-degree part of gλµ,ν and cλµ,ν . We present thoseresults in context of Matchings-Jack Conjecture of Goulden and Jackson.

We adapt the probabilistic concept of cumulants to the setup of a linearspace equipped with two multiplication structures. We present an algebraicformula which expresses a given nested product with respect to those twomultiplications as a sum of products of the cumulants. This formula leadsto some conclusions about the structure constants of Jack characters. Wealso show that our formula may be understood as an analogue of Leonov–Shiraev’s formula.

1

Streszczenie

W 1996 r. Goulden i Jackson wprowadzili rodzinę współczynników (cλµ,ν)indeksowaną trójkami partycji, która pojawia się w rozwinięciu w szeregpotęgowy pewnej sumy Cauchy’ego symetrycznych wielomianów Jacka J (α)

π .Goulden i Jackson przypuszali, że za współczynnikami cλµ,ν ukryta jest kom-binatoryka związana ze skojarzeniami. Postawiona przez nich hipoteza „OSkojarzeniach Jacka” pozostaje do dzisiaj otwarta.

Charaktery Jacka są uogólnieniem charakterów grup symetrycznych orazobiektami dualnymi do wielomianów Jacka. W rozprawie doktorskiej bada-my stałe strukturalne gλµ,ν charakterów Jacka. Są one uogólnieniem stałychstrukturalnych grup symetrycznych. Podajemy wzory na współczynniki naj-wyższych stopni wielomianów gλµ,ν i cλµ,ν . Prezentujemy te rezultaty w kon-tekście hipotezy „O Skojarzeniach Jacka”.

Adaptujemy probabilistyczne pojęcie kumulanty do struktury przestrzeniliniowej z dwoma mnożeniami. Prezentujemy formułę, która wyraża pewienmieszany iloczyn jako sumę kumulant. Znalezione wyrażenie prowadzi downiosków na temat stałych strukturalnych charakterów Jacka. Pokazu-jemy również, że nasza formuła może zostać uznana za odpowiednik formułyLeonova i Shiraeva.

2

Podziękowania

Autor jest niezmiernie wdzięczny wszystkim tym, którzy pomogli mu przyprzygotowaniu tej pracy.

Przede wszystkim pragnę złożyć podziękowania profesorowi PiotrowiŚniademu. Wprowadzenie do świata kombinatoryki algebraicznej, poświę-cony przy tym czas i energia, okazana pomoc przy kształtowaniu pracydoktorskiej, wszystko to trudno przecenić.

Jestem wdzięczny Maciejowi Dołędze za inspirujące rozmowy. DziękujęLeonowi Gondelmanowi za wsparcie i za pomoc przy korekcie tej pracy.Dziękuję Tomaszowi Godlewskiemu za wiele lat prawdziwej przyjaźni. Os-tatnie, ale nie mniejsze, wyrazy wdzięcznośći kieruje ku mojej wspaniałejrodzinie.

3

Contents

1 Introduction 5

2 Connection coefficients of Jack polynomials 152.1 Jack Polynomials and connection coefficients . . . . . . . . . 152.2 Matchings and maps . . . . . . . . . . . . . . . . . . . . . . . 212.3 Measures of non-orientability and

non-bipartiteness . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Structure constants of Jack characters 373.1 Jack characters and structure constants . . . . . . . . . . . . 373.2 The top-degree part of structure constants

and connection coefficients . . . . . . . . . . . . . . . . . . . . 403.3 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . 45

4 Algebras with two multiplications and their cumulants 574.1 Algebras with two multiplications . . . . . . . . . . . . . . . . 574.2 Analogue of Leonov–Shiryaev’s formula . . . . . . . . . . . . 634.3 Cumulant formula for Jack characters . . . . . . . . . . . . . 684.4 Proof of Proposition 4.9 . . . . . . . . . . . . . . . . . . . . . 714.5 Proof of Proposition 4.10 . . . . . . . . . . . . . . . . . . . . 78

Appendix A Top-degree parts in theMatchings-Jack Conjecture and the b-Conjecture 88

Bibliography 92

4

Chapter 1

Introduction

Jack polynomials

Jack polynomials(J

(α)π

)are a family of symmetric functions that depend

on a parameter α > 0 and is indexed by an integer partition π. They wereintroduced by Henry Jack in his seminal paper [Jac71]. For certain values ofα, Jack polynomials coincide with various well-known symmetric polynomi-als. For instance, up to multiplicative constants, Jack polynomials coincidewith Schur polynomials for α = 1; with the zonal polynomials, for α = 2;with the symplectic zonal polynomials, for α = 1/2; with the elementarysymmetric functions, for α = 0; and in some sense with the monomial sym-metric functions, for α = ∞. Since it has been shown that several resultsconcerning Schur and zonal polynomials can be generalized in a rather nat-ural way to Jack polynomials [Mac15, Section (VI.10)], Jack polynomialscan be viewed as a natural interpolation between several interesting familiesof symmetric functions.

Connections of Jack polynomials with various fields of mathematics andphysics were established: it turned out that they play a crucial role inunderstanding Ewens random permutations model [DH92], generalized β-ensembles and some statistical mechanics models [OO97], Selberg-type in-tegrals [Kan93], certain random partition models [Ker00], and some prob-lems of the algebraic geometry [Nak96, Oko03], among many others. Betterunderstanding of Jack polynomials is also very desirable in the context ofgeneralized β-ensembles and their discrete counterpart model [OO97]. Jackpolynomials are a special case of the Macdonald polynomials [Sta89, Mac15].

5

Connection coefficients

In 1996 Goulden and Jackson [GJ96] introduced two families of coefficients(cλµ,ν) and (hλµ,ν) depending on the parameter α and indexed by triples ofpartitions which arise in the power sum expansion of some Cauchy sum forJack symmetric functions, Section 2.1.3 explains some details more in depth.As Jack polynomials can be viewed as a natural interpolation between sev-eral interesting families of symmetric functions, so the coefficients (cλµ,ν) canbe viewed as an interpolation between the structure constants of the classalgebra and the double coset algebra.

Combinatorics of maps and matchings

A map [LZ04] is classically defined as a connected graph G (possibly, withmultiple edges) drawn on a surface Σ, i.e. a compact connected 2-dimensionalmanifold without boundary. The notion of a map is well established in enu-merative combinatorics. A vertex two-coloured map is called bipartite if eachedge connects vertices of different colors. For such a map we can canonicallyassign three partitions describing the distributions of both kinds of verticesand the distribution of the numbers of edges in the faces.

Matchings (also known as perfect matchings and pair partitions) areanother class of combinatorial objects appearing naturally in enumerativecombinatorics. There is a close relation between them and maps, see Sec-tion 2.2.3.

The Matchings-Jack Conjecture

Goulden and Jackson [GJ96] observed that some specialisations of the afore-mentioned coefficients cλµ,ν and hλµ,ν may be interpreted in terms of matchingsand maps respectively. Moreover, they observed that both coefficients seemto be polynomials in the variable β := α− 1 with non-negative integer coef-ficients. This supposition was expressed in two conjectures known today asthe Matchings-Jack Conjecture and the b-Conjecture. After more then 20years both still remain open.

Goulden and Jackson in their Matchings-Jack Conjecture suggested thatthe quantity cλπ,σ can be expressed as

cλπ,σ(β) =∑δ

βwtλ(δ),

where wtλ is some hypothetical combinatorial statistic and the sum runsover some special set of matchings. There are two easy specialisations of cλπ,σ

6

which indeed may be expressed in terms of matchings. Those specialisationscoincide with the connection coefficients of two commutative subalgebras ofthe group algebra of the symmetric group: the class algebra and the doublecoset algebra.

Except for some special cases there are no closed formulas for the co-efficients cλπ,σ. Bédard and Goupil [BG92] found a formula for c(n)

π,σ in therestricted case. Goulden and Jackson [GJ92] gave a bijective proof of thisresult. Goupil and Schaeffer [GS98] provide some formulas for c(n)

π,σ in thegeneral case. Morales and Vassilieva [MV13, Vas15] found closed formulasfor the expansion of the generating series of c(n)

π,σ using bijective methods.There are also some other results about the coefficients cλπ,σ [Bia05, Irv06].

Parallel to the studies on the form of the coefficients cλπ,σ, the suitablestatistic wtλ is being sought. Goulden and Jackson constructed some statis-tics wtλ for λ = [1n] and λ = [2, 1n−1] and proved the conjecture in thosecases [GJ96]. Later on, the Matchings-Jack Conjecture has been proved[KV16] in the case π = σ = (n) of the partitions with exactly one part.Recently, this result was strengthened [KVP18] to the case when one of thethree partitions is equal to (n) (under some assumptions).

The question of existence of the suitable statistic wtλ still remains open.However, it seems that the appropriate candidate η for a similar statisticwhich appears in the b-Conjecture was found [La 09] as a measure of non-orientability on a class of rooted maps.

The Matchings-Jack Conjecture and the b-Conjecture are ones of ma-jor open questions in enumerative algebraic combinatoric. They relate as-pects like: symmetric functions, the representation theory of the symmetricgroups, combinatorics of maps and matchings.

The first result

It was proved [DF16] that cλπ,σ are polynomials in the variable β := α−1 anda satisfactory bound on their degrees was given. We investigate the leadingcoefficient of the polynomials cλπ,σ. More precisely, in Theorem 2.5 we givea sufficient and necessary condition for the polynomial cλπ,σ to achive thebound on its degree given in [DF16]. We show that the leading coefficientof such cλπ,σ is a positive integer.

We present our result in the context of Matchings-Jack Conjecture ofGoulden and Jackson. Indeed, in Section 2.3.4 we construct the statisticstatη on a special class of matchings. We show that the leading coefficientof the polynomials cλπ,σ coincides with the leading coefficient relevant to the

7

statistic statη.In fact, we adapt the statistic η of La Croix [La 09] to the case of lists of

maps. In some special cases it may be translated into the field of matchings,however generally significant difficulties appear. Such attempts have alreadybeen made [KVP18]; it seems that the difficulty increases with the generality.In Section 2.3.4 we present briefly the problem of transferring the statistic ηinto a satisfactory statistic which measures non-bipartiteness of a matching.

The approach we use to prove Theorem 2.5 is fundamentally differentfrom previous attempts of other authors to prove the Matchings-Jack Con-jecture and the b-Conjecture. We investigate the structure constants for Jackcharacters, which are kind of dual objects to Jack polynomials. Althoughour research allows us so far to recover only the leading coefficients of thestructure constants and the connection coefficients, in Section 4.3 we intro-duce a method linked with the theory of cumulants, which in the future maygive information about the remaining coefficients of the polynomials cλπ,σ.

Unnormalized Jack characters

By expanding Jack polynomial in the basis of power-sum symmetric func-tions:

J(α)λ =

∑µ

θ(α)µ (λ) pµ

we get coefficients θ(α)µ (λ) called unnormalized Jack characters. Jack char-

acters θ(α)µ provide a kind of dual information about the Jack polynomials.

The coefficients appearing in the expansion of a pointwise product oftwo unnormalized Jack characters in the unnormalized Jack character basiscoincide with the connection coefficients [DF16], namely

θ(α)π · θ(α)

σ =∑µ`n

cµπ,σθ(α)µ .

This observation encourages us to look more deeply into the field of connec-tion coefficients via the context of Jack characters.

This kind of dual approach may be traced back to the work of Kerov andOlshanski [KO94] on the characters of the symmetric groups. The usual wayof viewing the characters of the symmetric groups is to fix the representationλ and to consider the character as a function of the conjugacy class π. Kerovand Olshanski suggested to do roughly the opposite. This idea was adaptedby Lassalle [Las08, Las09] to the framework of Jack characters.

8

Normalized Jack characters

In order for this dual approach to be successful one has to choose the mostconvenient normalization constants. We define (normalized) Jack charactersChπ as

Chπ(λ) :=

1√α

|π|+`(π)(|λ| − |π|+m1(π)

m1(π)

)zπ θ

(α)π∪1|λ|−|π|(λ) if |λ| ≥ |π|,

0 if |λ| < |π|,

where zπ is the standard numerical factor, and ∪ denotes concatenationof two partitions, see Section 2.1.1. At the first glance this choice of thenormalization factors may be confusing, however it is the appropriate one ifwe look for the asymptotic behaviour of Jack characters or for a convenientalternative description of them. It is worth mentioning that the characterChπ is a function on the set of all Young diagrams, which turns out to be apowerful tool of this approach.

We would like to notice that there are conjectures similar to the b-Conjecture and the Matchings-Jack Conjecture which involves objects dualto Jack polynomials. Lassalle proved that the characters Chπ are poly-nomials in terms of multirectangular coordinates and conjectured that thecoefficients of this expression are positive integers and posses some inter-pretation in terms of free cumulants [Las08, Las09]. Dołęga, Féray andŚniady conjectured that any given Jack character Chπ may be expressed asa weighted sum of some simple functions indexed by maps [DFS13].

Structure constants of symmetric groups

Ivanov and Kerov [IK99] established the notion of partial permutations insome suitable inverse limit of the symmetric group algebras C[S (n)]. For agiven partition they were adding an appropriate number of units to obtainthe partition of a relevant size. They observed that by a simple normal-ization of the conjugacy classes Aπ,n of the symmetric groups S (n) thefollowing convolution formula

Aπ,n ·Aσ,n =∑µ

gµπ,σAµ,n

holds for any sufficiently large n. The integers gµπ,σ appearing in the formulaabove are independent of the size of the group S (n). The group of finitepermutations acts naturally on the inverse limit of the semigroups of par-tial permutations. The numbers gµπ,σ arise as the multiplication structureconstants in the algebra of orbits of this action.

9

Structure constants

Structure constants gµπ,σ of Jack characters are defined by the expansion ofthe pointwise product of two Jack characters in the basis of Jack characters:

Chπ ·Chσ =∑µ

gµπ,σ(δ) Chµ

with the parameter δ :=√α − 1√

α. Explicit motivation for studying such

quantities comes from the special choice of the deformation parameter δ = 0,when Jack polynomials coincide with Schur polynomials. In this case, Frobe-nius duality ensures that the structure constants gµπ,σ(0) coincide with thestructure constants gµπ,σ for the symmetric groups introduced by Ivanov andKerov. It timidly suggests existence of some deformation of the symmetricgroup algebra C[S (n)] depending on the deformation parameter δ in whichgµπ,σ(δ) are the structure constants for the hypothetical conjugacy class in-dicators.

It has been observed [Śn16] that the quantities gµπ,σ(δ) seem to be poly-nomials with non-negative integer coefficients. Since the notion of the con-nection coefficients cµπ,σ and the notion of the structure constants gµπ,σ areclosely related, one can get an impression of triviality of this remark. Indeed,the structure constants gµπ,σ are some weighted decompositions of the con-nection coefficients [DF16] (see (3.3)) thus the polynomiality of both familiesis equivalent. The conjectures that the coefficients of these two polynomialsare non-negative integers seem to be closely related, however they are notequivalent.

The second result

In Theorem 3.3 we give a necessary and sufficient condition for the polyno-mial gλπ,σ to achieve the maximal degree provided by the bound of Dołęgaand Féray [DF16], we also show that the leading coefficient is a positiveinteger and we present it in terms of oriented maps. This is the key tool inthe proof of Theorem 2.5.

We were looking for combinatorial objects which may enumerate the co-efficients of the polynomials gλµ,ν . There was some evidence that there is acombinatorics of maps hidden behind them. Firstly we found a combinato-rial formula for the top-degree part of the character Chπ, see Proposition 3.9.This formula expresses

[Atop]Chπ(λ) in terms of injective embeddings of a

graph Gπ assigned to the character Chπ into the Young diagram λ. Thisformula brought us closer to discovering good candidates for the top-degree

10

part of structure constants, which was one of the difficulties. It turned outthat performing the “hands-shaking procedure”, see Section 3.3.3, providessuch candidates.

In order to prove that those candidates for the top-degree part of thestructure constants gµπ,σ are truly them we adapted the probabilistic conceptof cumulants to the setup of Jack characters. We present this approach inSection 4.3. Later on, we found a more direct proof of Theorem 3.3.

It is worth mentioning that the polynomiality of cµπ,σ and the bound on itsdegree have been proved by Dołęga and Féray [DF16] via the polynomialityand the bound on the degree of gµπ,σ. As understanding of the combinatoricsof Jack characters may lead to a better understanding of Jack polynomialsthemselves, so the combinatorial formulas for the structure constants maylead to some combinatorial formulas for the connection coefficients. Ourresult may be seen as an evidence for this general statement.

Cumulants

One of the classical problems in the probability theory is to describe the jointdistribution of a family (Xi) of random variables in the most convenient way.The most common solution of this problem is to use the family of moments,i.e. the expected values of products of the form E (Xi1 · · ·Xil). It has beenobserved that in many problems it is more convenient to make use of thecumulants [Hal81, Fis28], defined as the coefficients of the expansion ofthe logarithm of the multidimensional Laplace transform around zero. Forexample, the Gaussian distribution may be characterized by the vanishingof all cumulants except the first two (i.e. , other than mean and variance).

Cumulants allow also a combinatorial description. One can show thatthe definition of cumulants is equivalent to the following system of equations,called the moment-cumulant formula:

E (X1 · · ·Xn) =∑ν

∏b∈ν

κ (Xi : i ∈ b)

which should hold for any choice of the random variables X1, . . . , Xn whosemoments are all finite.

Generalized cumulants

The notion of cumulants was established also in a more general probabilisticsetup. One may consider the conditional expected value defined as a unitallinear map E : A −→ B between two commutative unital algebras and defineconditional cumulants similarly to the classical ones.

11

The notion of cumulants was transfered into the field of noncommuta-tive probability theory, where the noncommutative expectation is defined asa unital linear map φ on a unital algebra A [Spe94, Spe98]. Usually someother conditions, such as the bimodule map property, are required. RolandSpeicher introduced the free cumulant functional [Spe94] in the free prob-ability theory. It is related to the lattice of noncrossing partitions of theset [n] in the same way as the classic cumulant functional is related to thelattice of all partitions of that set.

Leonov–Shiryaev’s formula

In 1959 Leonov and Shiryaev [LS59] presented a formula for a cumulant ofproducts of random variables:

κ(

(X1,1 · · ·Xk1,1) , . . . , (X1,n · · ·Xkn,n))

in terms of simple cumulants. This formula was first proved by Leonov andShiryaev [LS59], a more direct proof was given by Speed [Spe83]. The tech-nique of Leonov and Shiryaev was used in many situations [SSR88, Leh04]and was further developed in other papers: Krawczyk and Speicher [KS00,MST07] found the free analogue of the formula; the formula was furthergeneralized to the partial cumulants [NS06].

Nested cumulants

We investigate the following particular case of the conditional cumulants.We assume that A is a linear space equipped with two commutative multipli-cation structures, which correspond to two products: · and ∗. Together witheach multiplication A forms a commutative algebra. We call such structurean algebra with two multiplications. As a mapping E we take the identityid : (A, ·) −→ (A, ∗).

In this case the cumulants measure the discrepancy between these twomultiplication structures on A. This situation arises naturally in manybranches of algebraic combinatorics, for example in the case of Macdonaldcumulants [Doł17a, Doł17b] and cumulants of Jack characters [DF17, Śn16].

Since the mapping E is the identity, we can define cumulants of cumu-lants and further compositions of them. The terminology of cumulants ofcumulants was introduced in [Spe83] and further developed in [Leh13] (calledthere nested cumulants) in a slightly different situation of an inclusion of al-gebras C ⊆ B ⊆ A and conditional expectations A E1−→ B E2−→ C.

12

The third result

In Theorem 4.6 we present an algebraic formula for a nested product whichinvolves two multiplications on the linear space A as a sum of ∗-products ofcumulants:(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑F∈F(A)

(−1)wF κF ,

where the sum runs over reduced forests with leaves labelled by elementsof the algebra A and satisfying some additional properties. Such forestsare assigned to the nested cumulants and are markers of the way they arenested.

A simple reformulation of our formula, see Theorem 4.16, may be seenas an analogue of Leonov–Shiryaev’s formula. Indeed, Leonov–Shiryaev’sformula relates a cumulant of products with some products of cumulants.In the framework of an algebra with two multiplications we can define twotypes of cumulants according to each multiplication separately. For each ofthem we have Leonov–Shiryaev’s formula. In Theorem 4.16 we present thethird formula, which is a mix of those two.

With the pointwise product and the, so called, disjoint product Jackcharacters span an algebra with two multiplications. Observe that in thiscontext our formula expresses the pointwise product of two Jack charactersas a sum of disjoint products of cumulants. Together with the approximatefactorisation property of cumulants [Śn16] our formula turns out to be a toolfor capturing the structure constants gµπ,σ, see Section 4.3.

At the initial stage of the research, this formula was an indispensable ele-ment in the proof of Theorem 3.3. Later on, we found a more direct proof ofTheorem 3.3 based on the formula on

[Atop]Chπ(λ) for all Young diagrams

λ, see Proposition 3.16. Originally we used a formula for[Atop]Chπ(λ) just

for one-row Young diagrams. This phenomenon suggests that the cumulantformula given in Theorem 4.6 may be useful while looking for combinatorialformulas for subdominant parts of the structure constants gµπ,σ. We believethat in the future the cumulant approach will bring new results about thestructure constants and indirectly about the connection coefficients from theMatchings-Jack Conjecture of Goulden and Jackson.

Structure of dissertation

The three next chapters of the dissertation are basically devoted to provingand discussing the three aforementioned main results.

13

Chapter 2 introduces the notions of Jack polynomials (J (α)π ) and their

connection coefficients cµπ,σ. In Section 2.1.5 we present the Matchings-JackConjecture. Theorem 2.5 presents the leading coefficient of the connectioncoefficients in the context of Matchings-Jack Conjecture of Goulden andJackson. In Section 2.2 we introduce the terminology of maps and we inves-tigate relations between maps and matchings. In Section 2.3 we present ameasure of non-orientability in the context of b-Conjecture. We present theproblem of transferring it into the satisfactory statistic which measures non-bipartiteness of a matching. We discuss recent results of Dołęga [Doł17c]about the top-degree part in b-Conjecture. Later, in Appendix A, we showthat our result about the top-degree part in the Matchings-Jack Conjecturepresented in Theorem 2.5 is equivalent to the result of Dołęga.

Chapter 3 is devoted to the dual approach, i.e. Jack characters Chπand their structure constants gµπ,σ. In Section 3.1 we show the relationbetween the structure constants gµπ,σ for Jack characters and the connectioncoefficients cµπ,σ for Jack symmetric functions. We give a formula for thetop-degree part of gµπ,σ and translate this result into the field of connectioncoefficients cµπ,σ. We prove this formula in Section 3.3.

Chapter 4 introduces the notion of cumulants in the classical probabilitytheory and adapts this probabilistic concept to the setup of algebras withtwo multiplications. Theorem 4.6 gives an algebraic formula which involvesthose two multiplications as a sum of products of cumulants. In Section 4.2we present this formula as an analogue of Leonov–Shiraev’s formula. Wefinish with some conclusions about the structure constants of Jack characterspresented in Section 4.3.

14

Chapter 2

Connection coefficients ofJack polynomials

2.1. Jack Polynomials and connection coefficientsIn this section we introduce Jack polynomials and their connection coeffi-cients. Theorem 2.5 presents the leading coefficient of connection coefficientsin the context of Matchings-Jack Conjecture of Goulden and Jackson.

2.1.1. Partitions

A partition λ of n (denoted by λ ` n) is a non-increasing list (λ1, . . . , λl) ofpositive integers of sum equal to n. Number n is called the size of λ and isdenoted by |λ|, the number l is the length of the partition, denoted by ` (λ).Finally,

mi(λ) :=∣∣{k : λk = i}

∣∣,is the multiplicity of i ≥ 1 in the partition λ.

There are many orders on the set of partitions. Beside the one shown inDefinition 2.3 we introduce the dominance order. We say that λ ≤ µ if andonly if ∑

i≤jλi ≤

∑i≤j

µi

holds for any positive integer j.For given two partitions λ and µ we construct their concatenation (de-

noted λ ∪ µ) by merging all parts from λ and µ and ordering them in adecreasing fashion.

15

2.1.2. Jack polynomials

Jack polynomials(J

(α)π

)are a family of symmetric functions that depend

on a parameter α > 0 and is indexed by an integer partition π. They wereintroduced by Henry Jack in his seminal paper [Jac71]. Jack polynomialscan be viewed as a natural interpolation between several interesting familiesof symmetric functions. For instance, up to multiplicative constants, Jackpolynomials coincide with Schur polynomials for α = 1; with the zonalpolynomials, for α = 2; with the symplectic zonal polynomials, for α = 1/2;with the elementary symmetric functions, for α = 0; and in some sense withthe monomial symmetric functions, for α = ∞. There are many ways todefine Jack polynomials, we present one of them [Mac15, Section VI.10].

Let us consider the vector space ΛQ(α) of the symmetric functions [DKB66]over the field of rational functions Q(α) and its basis (pλ)λ of power-sumsymmetric functions, i.e. the symmetric functions given by

pλ(x) =∏i

pλi(x), pk(x) = xk1 + xk2 + · · · .

The following scalar product on ΛQ(α) is defined on the power-sum basisby the formula

〈pλ, pµ〉α := α`(λ)zλδλ,µ,

wherezλ =

∏i

imi(λ) mi(λ)!

and further extended by bilinearity. This is a classical deformation of theHall inner product, which corresponds to α =1 [Jac71].

Jack polynomials are the only family of symmetric functions(J

(α)π

)which satisfies the following three criteria:

1. J (α)λ =

∑µ≤λ a

λµmµ, where aλµ ∈ Q[α],

2.[m1|λ|

]J

(α)λ := aλ1|λ| = |λ|!,

3. 〈J (α)λ , J

(α)µ 〉α = 0 for λ 6= µ,

where mλ denotes the monomial symmetric function associated with λ.

16

2.1.3. Connection coefficients for Jack symmetric functions

Goulden and Jackson [GJ96] defined two families of coefficients(cλπ,σ

)and(

hλπ,σ

)depending implicitly on the deformation parameter α and indexed

by triples of integer partitions π, σ, λ ` n of the same integer n. Thesecoefficients are given by expansions of the left-hand sides in terms of thepower-sum symmetric functions:

∑θ∈P

1〈Jθ, Jθ〉α

J(α)θ (x)J (α)

θ (y)J (α)θ (z)t|θ| =

∑n≥1

tn∑

λ,π,σ`n

cλπ,σα`(λ) z

−1λ pπ(x)pσ(y)pλ(z), (2.1)

and

αt∂

∂tlog

∑θ∈P

1〈Jθ, Jθ〉α

J(α)θ (x)J (α)

θ (y)J (α)θ (z)t|θ|

=

∑n≥1

tn∑

λ,π,σ`nhλπ,σpπ(x)pσ(y)pλ(z), (2.2)

see [GJ96, Equations (1),(5) and Equations (2),(4)].Dołęga and Féray showed that the connection coefficients (cλπ,σ) are poly-

nomials in the variable β := α− 1 with rational coefficients and proved thefollowing upper bound on the degrees of these polynomials [DF16, Proposi-tion B.2.]:

degβ cλπ,σ ≤ d (π, σ;λ) , (2.3)

where

d (π, σ;λ) :=(|π| − `(π)

)+(|σ| − `(σ)

)−(|λ| − `(λ)

).

One may wonder of the use of the new variable β, but this shift seems tobe the adequate one in order to look at the connection coefficients from thecombinatorial point of view.

2.1.4. Matchings

We present the well established terminology of matching given in [GJ96].For a given integer n we consider the following set

Nn ={

1, 1, . . . , n, n}.

17

We denote by Fn the set of all matchings (partitions on two-elements sets)on Nn. For matchings δ1, δ2, . . . ∈ Fn we denote by G(δ1, δ2, . . .) the multi-graph with the vertex set Nn whose edges are formed by the pairs inδ1, δ2, . . .. For given matchings δ1, δ2 the corresponding graph G(δ1, δ2) con-sists of disjoint even cycles, since each vertex has degree 2 and around eachcycle the edges alternate between δ1 and δ2. Denote by Λ(δ1, δ2) the parti-tion of n which specifies halves the lengths of the cycles in G(δ1, δ2). Moregenerally, denote by Λ(δ1, . . . , δs) the partition of n which specifies halves ofthe number of vertices in each connected component of G(δ1, δ2, . . .) (it isan easy observation that such numbers form a partition of n).

We call the sets {1, . . . , n} and {1, . . . , n} classes of Nn. A pair in amatching is called a between-class pair if it contains elements of differentclasses. A matching δ in which every pair is a between-class pair is called abipartite matching (in this case G(δ) is a bipartite graph on the vertex-setsgiven by the two classes of Nn).

We introduce two specific bipartite matchings in the set Fn. First, let

ε :={{1, 1}, . . . , {n, n}

};

second, for a given partition µ ` n, let

δλ :={{1, 2}, {2, 3}, . . . , {λ1 − 1, λ1}, {λ1, 1},

{λ1 + 1, ˆλ1 + 2}, . . . , {λ1 + λ1 − 1, ˆλ1 + λ1}, {λ1 + λ1, ˆλ1 + 1}, . . .},

see Figure 2.1. Observe that both matchings: ε and δλ are bipartite andΛ(ε, δλ) = λ.

2.1.5. Matchings-Jack Conjecture

Definition 2.1. For given three partitions π, σ, λ ` n, we denote by Gλπ,σ theset of all matchings δ ∈ Fn, for which Λ(δ, ε) = π and Λ(δ, δλ) = σ.

Goulden and Jackson observed that the specializations of cλπ,σ(β) forβ ∈ {0, 1} may be expressed in terms of matchings, namely

cλπ,σ(0) =∣∣∣{δ ∈ Gλπ,σ : δ is bipartite

}∣∣∣,cλπ,σ(1) =

∣∣∣{δ ∈ Gλπ,σ}∣∣∣.In fact, those specialisations coincide with the connection coefficients of twocommutative subalgebras of the group algebra of the symmetric group: the

18

1

2

3

4

1

2

3

4 5

6

5

6

7

8

7

8

λ = (4, 2, 2)

δλ :={{1, 2}, {2, 3}, {3, 4}, {4, 1}︸ ︷︷ ︸

λ1=4

, {5, 6}, {6, 5}︸ ︷︷ ︸λ2=2

, {7, 8}, {8, 7}︸ ︷︷ ︸λ3=2

}

Figure 2.1 – An example of matchings ε (dotted line) and δλ (continuousline) for λ = (4, 2, 2). Observe that both matchings: ε and δλ are bipartiteand Λ(ε, δλ) = λ.

class algebra and the double coset algebra (β = 0 and β = 1 respectively)[HSS92].

Based on this observation Goulden and Jackson conjectured that thefamily

(cλπ,σ

)of polynomials may have a combinatorial interpretation. The

conjecture is known as the Matchings-Jack Conjecture.

Conjecture 2.2 (Matchings-Jack Conjecture). For any partitions π, σ, λ `n the quantity cλπ,σ can be expressed as

cλπ,σ(β) =∑

δ∈Gλπ,σ

βwtλ(δ),

where wtλ : Gλπ,σ −→ N0 is some hypothetical combinatorial statistic, whichvanishes if and only if δ is bipartite.

Clearly, it seems that the statistic wtλ should be a marker of non-bipartiteness for matchings. Matchings-Jack Conjecture remains still openin the general case, however some special cases have been settled. Gouldenand Jackson constructed some statistics wtλ for λ = [1n] and λ = [2, 1n−1]and proved the conjecture in those cases [GJ96]. Later on, the Matchings-Jack Conjecture has been proved by Kanunnikov and Vassiliveva [KV16] inthe case π = σ = (n) of the partitions with exactly one part. Recently, in

19

Figure 2.2 – A pair of partitions λ = (4, 3, 1, 1) and µ = (5, 3, 1) presentedas Young diagrams. Partition λ is sub-partition of µ; indeed, each part of µis given as a sum of different parts of λ.

a joint paper with Promyslov [KVP18], they proved the conjecture in thecase when one of the three partitions is equal to (n). They made use of themeasure of non-orientability θ defined by La Croix in his PhD thesis [La 09].The measure of non-orientability θ is a statistic defined on a class of rootedmaps. In some special cases it may be translated into the field of matchings,however generally significant difficulties appear. We also shall use the samestatistic.

2.1.6. The first result

In this dissertation we give a necessary and sufficient condition for the poly-nomial cλπ,σ to achieve the maximal degree given by (2.3). Moreover, we showthat the leading coefficient of cλπ,σ of this maximal degree is a non-negativeinteger and we present it in the context of Matchings-Jack Conjecture.Definition 2.3. Consider two integer partitions λ and µ of the same integern, let k = `(λ) and m = `(µ) be the lengths of the partitions. We say thatλ is a subpartition of µ (denoted λ � µ) if there exists a set-partition ν of[k], such that

µi =∑j∈νi

λj

for any i ∈ [m], see Figure 2.2. We denote λ ≺ µ if λ � µ and λ 6= µ.

Definition 2.4. For given partitions π, σ, λ, µ ` n, we denote by Gλ;µπ,σ the set

of all matchings δ ∈ Gλπ,σ which are µ-connected, i.e. Λ(δ, ε, δλ) = µ.

The class Gλπ,σ splits naturally into the classes Gλ;µπ,σ , namely

Gλπ,σ =⊔

µ:λ�µGλ;µπ,σ .

20

Contrary to previous works on the Matchings-Jack Conjecture we do notattempt to define the statistic wtλ on Gλπ,σ for a particular class of partitionsλ, π or σ. We define the statistic "statη" on the class Gλ;λ

π,σ .

Theorem 2.5 (The first result). For any triple of partitions π, σ, λ ` nthe corresponding polynomial cλπ,σ(β) achieves the upper bound on the degreegiven in (2.3) if and only if π and σ are sub-partitions of µ. For suchpartitions, the leading coefficient of cλπ,σ(β) may be expressed in two differentmanners:[

βd(π,σ;λ)]cλπ,σ =

∣∣∣δ ∈ Gλ;λπ,σ : δ is unhandled

∣∣∣ =∑ν:ν�λ

zλzν

∣∣∣δ ∈ Gν;λπ,σ : δ is bipartite

∣∣∣,for notion of unhandled matchings see Definition 2.28. Moreover, thereexists a statistic statη : Gλ;λ

π,σ −→ N0, which satisfies[βd(π,σ;λ)

]cλπ,σ =

[βd(π,σ;λ)

] ∑δ∈Gλ;λ

π,σ

βstatη(δ)

and for δ ∈ Gλ;λπ,σ the statistic statη(δ) vanishes if and only if δ is bipartite.

2.2. Matchings and mapsIn this section we shall present the notion of maps. We also show relationsbetween them and matchings.

2.2.1. Maps

In the literature a map [LZ04] is classically defined as a connected graphG (possibly, with multiple edges) drawn on a surface Σ, i.e. a compactconnected 2-dimensional manifold without boundary. We assume that acollection of faces (i.e. Σ \ E) is homeomorphic to a collection of open discs.A choice of an edge-side and one of its endpoints is called a root of the map,see Figure 2.3. A map together with a choice of a root is called a rootedmap.

A vertex two-coloured map is called bipartite if each edge connects ver-tices of different colors; for simplicity we set that there are white and blackvertices, we denote by W (B) the set of white (black) vertices. By conven-tion, from a rooted bipartite map we require that the rooted vertex is black.

21

ΛF (M) = (4, 2, 2)ΛW (M) = (6, 2)ΛB (M) = (2, 2, 2, 2)

Figure 2.3 – Example of a rooted bipartite mapM on a projective plane. Theleft side of the square should be glued to the right side, as well as bottom totop, as indicated by the arrows. We present also the face, white and blackvertex distributions.

Figure 2.3 presents an example of a rooted bipartite map M . For a givenbipartite map M with n edges we establish two integer partitions of n:

ΛW (M) and ΛB (M) ,

given by the degrees of white/black vertices, For such a map we assign alsothe third partition

ΛF (M)of n, which describes the face structure ofM ; it is specified by reading halvesof the numbers of edges fencing each face (since the map M is bipartite, foreach face there is an even number of edges adjacent to the face). Thepartition ΛF (M) is called the face-type of the map M .Definition 2.6. For three given partitions π, σ, λ ` n we denote by M•π,σ theset of all bipartite, rooted maps M with n edges, for which ΛW(M) = πand ΛB(M) = σ. Moreover we denote by Mλ

π,σ the set of all such a maps Mwhich additionally have the face-type λ, i.e. ΛF (M) = λ, see Figure 2.3.

Due to the nature of our result we extend this definition slightly, namelywe waive the assumption of connectedness in the definition of a map. Thereare two natural ways to generalize the notion of connected maps to non-connected ones: either we consider lists of connected maps or we considercollections of them.

2.2.2. Lists and collections of maps

Definition 2.7. Let µ = (µ1, . . . , µk) be a partition of an integer n. A list ofmaps (M1, . . . ,Mk) is called a µ-list of maps if the map Mi has µi edges for

22

µ = (4, 4)ΛF (M) = (4, 2, 2)ΛW (M) = (4, 4)ΛB (M) = (2, 2, 2, 2)

µ1 = 4 µ2 = 421

Figure 2.4 – Example of a rooted, bipartite µ-list of maps for µ = (4, 4).The first map is drawn on a torus, the second one on a projective plane.We present also the face, white vertices and black vertices distributions. Byerasing the roots and the numbering of the connected components we obtaina bipartite µ-collection of maps.

each i ∈ [k]. We say that such a list is rooted, respectively bipartite if eachmap Mi is so. For a bipartite µ-list of maps we associate three partitionsdescribing the black vertex, the white vertex and the face structures

ΛW(M) :=k⋃i=1

ΛW(Mi), ΛB(M) :=k⋃i=1

ΛB(Mi), ΛF (M) :=k⋃i=1

ΛF (Mi),

where⋃

denotes the concatenation of partitions.Definition 2.8 (Extension of Definition 2.6). For given partitions π, σ, µ ` n,we denote by M•;µπ,σ the set of all bipartite rooted µ-lists of maps M whichsatisfy

ΛW(M) = π and ΛB(M) = σ,

see Figure 2.4. Moreover, for a given partition λ ` n we denote by Mλ;µπ,σ the

set of all µ-lists of mapsM ∈M•;µπ,σ which have face-type λ, i.e. ΛF (M) = λ.Definition 2.9. Let µ = (µ1, . . . , µk) be a partition of an integer n. A setof maps {M1, . . . ,Mk} is called a µ-collection of maps if the map Mi has µiedges for each i ∈ [k]. We say that such a collection is rooted or bipartiteif each map Mi is so. For such a collection of maps we associate threepartitions describing black, white and face structures as in Definition 2.7.

23

Roughly speaking, a µ-collection of maps could be created from a µ-listof maps by erasing the numbering of the connected components, i.e. theorder on the connected components (see Figure 2.4).

2.2.3. Matchings and maps

Matching and maps are closely related notions. Roughly speaking, a bipar-tite matching can be treated as a (possibly non-connected) bipartite mapwith rooted and numbered faces. We shall discuss relations between match-ings and rooted list of maps with the same face, black vertices and whitevertices distribution.Definition 2.10. Consider two partitions λ, µ ` n. We say that a bipartiteµ-collection M of maps with the face distribution given by λ has rooted andnumbered faces if all faces of M are rooted (i.e. on each face there is onemarked edge-side) and the face labelled by the number i is surrounded by2λi edges, for each i, see Figure 2.5. The set of such collections of mapswith the face, black vertices and white vertices distributions given by thepartitions λ, π, σ ` n is denoted by M

(Gλ;µπ,σ

).

Remark 2.11. Observe that rooting a face is nothing else but choosing oneof the face corners adjacent to some black vertex and orienting the face.Through a map (or a list/a collection of maps) with rooted faces we canunderstand a map with oriented faces and chosen black corners for each ofthe faces, see Figure 2.5. Similarly, rooting a map is choosing one corner ofa black vertex and orienting the face adjacent to this corner.

We consider four partitions: π, σ, λ, µ ` n. To a given matching δ ∈ Gλ;µπ,σ

we associate a bipartite µ-collection Mδ ∈ M(Gλ;µπ,σ

)given by the following

procedure.

1. The matchings ε and δλ determine the polygons with the vertices la-belled by Nn, see Figure 2.1. We take theirs duals, i.e. the polygonswith the edges labelled by Nn, see Figure 2.6. The consecutive poly-gons have 2λ1, 2λ2, . . . edges. Observe that the parts of ε (respectivelyδλ) can be identified with the black (respectively white) vertices as itis shown on Figure 2.6;

2. The matching δ determines the unique way of gluing the edges of thepolygons in such a way that black (white) vertices are glued with black(white) ones. Figure 2.7 presents such a gluing for the matching

δ ={{1, 6}, {1, 5}, {2, 8}, {2, 7}, {3, 7}, {3, 8}, {4, 6}, {4, 5}

}.

24

Figure 2.5 – Example of a bipartite (4, 4)-collection of maps with rootedfaces. By rooting faces we understand choosing one edge-side of each face(drawn as a black half-arrow going from a black vertex) or, equivalently,orienting each face (the rounded arrows) and choosing one black vertex foreach face (the red arrows).

Observe that the distribution of black (respectively white) vertices isgiven by Λ(δ, ε) (respectively by Λ(δ, δλ)). Moreover, µ = Λ(δ, ε, δλ).

3. Each face is canonically numbered by an integer s related to the poly-gon λs, i.e. the edge-sides of this face are labelled by the elements

s−1∑i=1

λi + 1, . . . ,s−1∑i=1

λi + λs,

s−1∑i=1

λi + 1, . . .s−1∑

i=1λi + λs.

Such a face is canonically rooted by selecting the edge-side labelled bythe number

∑s−1i=1 λi + 1, see Figure 2.7.

4. We remove the labelling by the elements from Nn.

Corollary 2.12. The procedure described above gives a bijection δ 7→Mδ be-tween the set of matchings Gλ;µ

π,σ and the set of collections of maps M(Gλ;µπ,σ

).

We compare the terminologies of matchings and maps in the table below.

25

1

2

3

4

1

2

3

4 5

6

5

6

7

8

7

8

Figure 2.6 – Duals of the polygons created by the matchings ε and δλ pre-sented on Figure 2.1. Black (respectively white) vertices of such polygonsare labelled by the elements of ε (respectively δλ), the edges by the elementsfrom N8.

2

77

8 8

23

3

64

45

61

511

6

451

2

3

δ ={{1, 6}, {1, 5}, {2, 8}, {2, 7}, {3, 7}, {3, 8}, {4, 6}, {4, 5}

}Figure 2.7 – Matching δ on the set N8 describes the way of gluing thesides of the polygons from Figure 2.6. Labels from N8 determine the wayof numbering and rooting faces of such a map (in general it could be acollection of maps), the roots (presented as half-arrows) correspond to thelabels 1, 5, 7. Numbers of faces are presented in black circles.

26

Matching δ µ-list of maps M

face-type λ ΛF (M)

distribution of black vertices Λ(δ, ε) ΛB(M)

distribution of white vertices Λ(δ, δλ) ΛW(M)

connected components Λ(δλ, ε, δ) µ

µ-collections of maps with given faces,black and white vertices distributionand with rooted and numbered faces

Gλ;µπ,σ M

(Gλ;µπ,σ

)

2.2.4. Matchings and lists of rooted maps

We showed that matchings are equivalent to collections of maps with rootedand numbered faces. However, collections of maps with rooted and num-bered connected components (i.e. lists of rooted maps) are much more nat-ural objects. We give a relation between those two ways of numbering androoting collections of maps. More precisely, we present a relation betweenthe set M

(Gλ;µπ,σ

)and the set Mλ;µ

π,σ .What is common for those two classes is the fact that by rooting and

numbering faces or connected components, the group of automorphisms be-comes trivial.Definition 2.13. For a given µ-collection of maps with rooted and num-bered faces M ∈M

(Gλ;µπ,σ

)we define the set R(M) of all numberings of the

connected components and rooting each of them in such a way that withrespect to them M becomes a µ-list of maps from Mλ;µ

π,σ . We call R(M)the set of components-labellings of M . For a given r ∈ R(M) we denote(M, r) ∈Mλ;µ

π,σ .Similarly, for a given µ-list of mapsM ∈Mλ;µ

π,σ we define the set L(M) ofall numberings of the faces and rooting each of them in such a way that Mbecomes an element fromM

(Gλ;µπ,σ

). We call L(M) the set of faces-labellings

of M . For a given l ∈ L(M) we denote (M, l) ∈M(Gλ;µπ,σ

).

Observation 2.14. Let us fix partitions π, σ, µ, λ ` n. For each M1 ∈M(Gλ;µπ,σ

)and M2 ∈Mλ;µ

π,σ we have∣∣∣R(M1)∣∣∣ = 2`(µ)zµ and

∣∣∣L(M1)∣∣∣ = 2`(λ)zλ.

Proof. Let us take M ∈ M(Gλ;µπ,σ

). There is

∏imi(µ)! ways of numbering

27

the connected components and∏i

(2i)mi(µ) ways of rooting each of them.

We may carry out a similar deduction for M ∈Mλ;µπ,σ .

Observation 2.15. For given partitions π, σ, µ, λ ` n we have

∣∣∣Gλ;µπ,σ

∣∣∣ =∣∣∣M (

Gλ;µπ,σ

) ∣∣∣ = zλ2`(λ)

zµ2`(µ)

∣∣∣Mλ;µπ,σ

∣∣∣.Proof. The first equation follows from Corollary 2.12. We investigate thesecond one. Each collection of maps from M

(Gλ;µπ,σ

)has rooted and num-

bered faces, each collection of maps from Mλ;µπ,σ has rooted and numbered

components. From each of them we can get a collection of maps which haverooted and numbered both: faces and components. The number of ways ofdoing it is given in Observation 2.14. We use the double counting methodand conclude the second equation.

2.2.5. Orientable maps and bipartite matchings

By an orientable map we understand a map which is drawn on an orientablesurface. An orientation of a map is given by orienting each face in sucha way, that the two edge-sides forming the same edge are oriented in theopposite way. We say that such an orientation of faces is coherent. Orient-ing any face is equivalent to orienting a map. Observe that a rooted mappossesses the canonical orientation given by the root, see Remark 2.11. Bya rooted orientable map we understand an orientable map together with theorientation given by the root, see Figure 2.8.Definition 2.16. We use the following notation:

Mλ;µπ,σ :=

{M ∈Mλ;µ

π,σ : M is orientable},

M•;µπ,σ :={M ∈M•;µπ,σ : M is orientable

},

Gλ;µπ,σ :=

{δ ∈ Gλ;µ

π,σ : δ is bipartite}.

The notion of bipartiteness of a matching is closely related to the notionof orientability.

Observation 2.17. For given partitions π, σ, µ, λ ` n, we have∣∣∣Gλ;µπ,σ

∣∣∣ = zλzµ

∣∣∣Mλ;µπ,σ

∣∣∣.

28

Figure 2.8 – Example of a rooted oriented map M drawn as a graph on atorus (on the left). There is the canonical orientation (grey arrows) givenby the root. We are going to present oriented maps in such a way thattheir orientation is consistent with the clockwise orientation of the page(grey arrows) or, equivalently, the counter-clockwise orientation around eachvertex (red arrows). The distinction between chosen orientations of the pageand the vertices may seem awkward. However, it is more convenient for thepurpose of Section 2.3.2. With this convention we can present the rootof a map (similarly roots of lists of maps) by an arrow going out from ablack vertex. Since M is oriented, it can be recovered from a graphicalrepresentation on the plane as a graph with a fixed cyclic order of outgoingedges around each vertex together with a choice of the root (on the right).

29

Proof. We identify a matching δ ∈ Gλ;µπ,σ with a collection of maps Mδ ∈

M(Gλ;µπ,σ

)with rooted and numbered faces by the procedure described in

Section 2.2.3. Observe that a bipartite matching corresponds to a collectionof oriented maps. Indeed, the orientations of faces given by the edge-sides:1, λ1 + 1, . . . are coherent. Observation 2.15 gives a relation between collec-tions of maps with rooted and numbered faces and collections of maps withrooted and numbered components (lists of maps). An analysis similar to theone given in Observation 2.15 convinces us that the quantity 2`(µ)∏

i imi(λ)

specifies the number of manners of rooting the faces in a coherent way and∏imi(λ)! specifies the number of manners of numbering the faces. On the

other hand, the quantity zµ2`(µ) is relevant for numbering and rooting theconnected components. We use the double counting method and concludethe statement.

2.3. Measures of non-orientability andnon-bipartitenessIn this section we shall present a measure of non-orientability η in the contextof b-Conjecture. We present a problem of transferring η into a satisfactorystatistic which measures non-bipartiteness of a matching. We discuss theresult of Dołęga [Doł17c] about the top-degree part in b-Conjecture.

2.3.1. The b-Conjecture

Equations (2.1) and (2.2) define two families of coefficients(cλµ,ν

)and

(hλµ,ν

).

Goulden and Jackson [GJ96] discussed some specialisations of the family(cλµ,ν

)and hypothetical combinatorial interpretations of the polynomials

cλµ,ν in terms of matchings known as the Matchings-Jack Conjecture, see Sec-tion 2.1.5. In the same paper they observed that specializations of hλπ,σ(β)for β = 0, 1 may be expressed in terms of rooted maps, namely

hλπ,σ(0) =∣∣∣{M ∈Mλ

π,σ : M is orientable}∣∣∣,

hλπ,σ(1) =∣∣∣{M ∈Mλ

π,σ

}∣∣∣.Based on this observation Goulden and Jackson conjectured that the fam-ily

(hλπ,σ

)of polynomials may have a combinatorial interpretation. The

conjecture is known as the b-Conjecture.

30

Conjecture 2.18 (b-Conjecture). For any partitions π, σ, λ ` n the quantityhλπ,σ can be expressed as

hλπ,σ(β) =∑

M∈Mλπ,σ

βη(M),

where η : Mλπ,σ −→ N0 is some hypothetical combinatorial statistic such that

η(M) = 0 if and only if M is orientable.

2.3.2. Root-deletion procedure and a measure of non-orientability

The statistic η from b-Conjecture should be a marker of non-orientabilityof maps. We shall present the definition of the measure of non-orientabilityintroduced by La Croix [La 09, Definition 4.1], which seems to be a goodcandidate for the hypothetical statistic conjectured by Goulden and Jackson.We adapt the statistic given by La Croix to the case of lists of maps.Definition 2.19 (Root-deletion procedure). Denote by e the root edge of themap M . By deleting e from M we create either a new map, or two newmaps. We give the canonical procedure of rooting it or them. By rooting amap we will understand choosing an oriented corner, see Figure 2.9. Denoteby c the root corner of M .

Suppose that M \ e is connected. Observe that c is contained in theunique oriented corner of M \ e, we define such an oriented corner as theroot of M \ e.

Suppose that M \ e has two connected components. One of them can berooted as above. Observe that the first corner in the root face ofM followingc is contained in the unique oriented corner of the second component ofM \ e, see Figure 2.9. We define such an oriented corner as the root of thiscomponent.Remark 2.20. The Root-deletion procedure is defined for all maps, not nec-essary bipartite. In particular, we do not require that the rooted vertex isblack.

We classify the root edges of maps. Let f be the number of faces of amap M with the root vertex e;

1. e is called a bridge if M \ e is not connected,

2. otherwise M \ e is connected and e is called

• a border if the number of faces in M \ e is equal to f − 1,

31

cc′

Figure 2.9 – The oriented corner c (red arrow) equivalent to the root (theblack arrow) of a map. The first corner in the root face of the map followingc is labelled by c′ (red arrow). By deleting the root edge the map splitsinto two new maps. The oriented corners c and c′ are contained in twooriented corners of the new maps. They give the roots of those maps (theblue arrows).

• a twisted edge if the number of faces in M \ e is equal to f ,• a handle if the number of faces in M \ e is equal to f + 1.

Remark 2.21. A leaf (i.e. an edge connecting a vertex of degree 1) is con-sidered as a bridge.Definition 2.22. [La 09, Definition 4.1] For a rooted map M , an invariantη(M) is defined inductively as follows.

1. If M has no edges then η(M) = 0.

2. Otherwise M has the root edge e,

• η(M) = η(M1) + η(M2) if e is a bridge, while M1 and M2 are theconnected components of M \ e,• η(M) = η(M \ e) if e is a border,• η(M) = η(M \ e) + 1 if e is a twisted edge,• if e is a handle, there exists a unique map M ′ with the root edgee′ constructed by twisting the edge e in M , in such a way that e′is a handle and the maps M \ e, M ′ \ e are equal. In this case werequire that{

η(M), η(M ′)}

= {η(M \ e), η(M \ e) + 1} .

32

At most one of the maps M , M ′ is orientable. For such a mapM we require η(M) = η(M \ e).

We call such an invariant a measure of non-orientability.Observe that the above definition introduces the whole family of mea-

sures of non-orientability η and among of them there is no canonical measureof non-orientability.Remark 2.23. For a given rooted map M

η (M) = 0 if and only if M is orientable.

Indeed, removing twisted edges or handles during the root-deletion proce-dure are the only possibilities of increasing the recursively-defined statistic η.An orientable map does not have any twisted edges (a map with a twistededge is embedded in a surface which contains the Möbius strip, hence isnonorientable). The recursive definition of η guarantees that removing han-dles from an orientable map does not increase the statistic η. Hence foran orientable map M , we have η(M) = 0. A reverse analysis or a simpleinduction on the number of edges provides the reverse implication.Definition 2.24. For a rooted µ-list of maps M = M1, . . .Mk we define ameasure of non-orientability η of M by

η (M) := η1 (M1) + · · ·+ ηk (Mk)

for any measures of non-orientability ηi from Definition 2.22.

2.3.3. Unhandled and unicellular maps

Definition 2.25. The rooted map M is called unhandled if by iterativelyperforming the root-deletion process (see Definition 2.19) it does not haveany handles. The map M is called unicellular if it has only one face.

From now on we fix one of measures of non-orientability η of the classof maps. Dołęga [Doł17c, Section 4] showed that for such a measure η thepolynomial Hη given by the sum

(Hη)λπ,σ :=∑

M∈Mλπ,σ

βη(M)

has degree at most equal to n + 1 − `(π) − `(σ) and the leading coefficientis enumerated by unhandled unicellular maps. In particular, (Hη)λπ,σ mayachieve this bound of the degree only if λ = (n). He also showed that the

33

aforementioned leading coefficient is also enumerated by oriented maps witharbitrary face-type, namely∣∣∣M ∈M•π,σ : M is orientable

∣∣∣ =∣∣∣M ∈M (n)

π,σ : M is unhandled∣∣∣.

In fact, there is an explicit bijection between those two families of maps.Dołęga proved [Doł17c, Theorem 1.4] that for the statistic η

h(n)π,σ(β) =

∑M∈M(n)

π,σ

βη(M)

holds true for β ∈ {−1, 0, 1}, moreover forM ∈M (n)π,σ the statistic η(M) = 0

vanishes if and only ifM is orientable; furthermore η(M) = n+1−`(π)−`(σ)if and only if M is unhandled and unicellular.

The result of Dołęga is easily transferable to the context of µ-lists ofmaps. Let us choose the measures of non-orientability ηi for i ∈ [k], k = `(µ),which form the measure η as it is described in Definition 2.24.

Lemma 2.26. For the statistic η, the polynomial (Hη)λ;•π,σ given by the sum

(Hη)λ;•π,σ :=

∑µ:λ�µ

(Hη)λ;µπ,σ (2.4)

where(Hη)λ;µ

π,σ (β) =∑

M∈Mλ;µπ,σ

βη(M) (2.5)

is of degree at most d(π, σ;λ). Moreover, a µ-lists of maps M contributesto the ground term if and only if M is a list of orientable maps. The µ-listsof maps M contributes to the leading coefficient if and only if M is a list ofunicellular and unhandled maps, in particular µ = λ.

Proof. Each M = (M1, . . . ,Mk) ∈ Mλ;µπ,σ decompose into a list of maps

Mi ∈Mλ|µiπ|µi ,σ|µi

for some partitions π|µi , σ|µi , λ|µi ` µi satisfying

k⋃i=1

π|µi = π,k⋃i=1

σ|µi = σ,k⋃i=1

λ|µi = λ.

We denote by Pµπ the set of lists of partitions(π|µ1 , . . . , π|µk

), where π|µi ` µi

andk⋃i=1

π|µi = π.

34

Observe, that (2.4) can be rewritten in such a way:

∑M∈Mλ;µ

π,σ

βη(M) =∑

(π1,...,πk)∈Pµπ(σ1,...,σk)∈Pµσ(λ1,...,λk)∈Pµ

λ

k∏i=1

∑M∈Mλi

πi,σi

βηi(M).

We use the result of Dołęga for each most right side sum separately. Eachsuch a sum has degree at most equal to µi + 1 − `(πi) − `(σi) and thetop-degree coefficient is enumerated by unhandled unicellular maps. Since

n+ `(µ)− `(π)− `(σ) =k∑i=1

(µi + 1− `(πi)− `(σi)

),

we conclude that (2.4) has degree at most equal to d(π, σ;λ) and the top-degree coefficient is enumerated by µ-lists of unhandled unicellular maps.

Corollary 2.27. For three given partitions π, σ, λ ` n we have∣∣∣M ∈M•;µπ,σ : M is orientable∣∣∣ =

∣∣∣M ∈Mµ;µπ,σ : M is unhandled

∣∣∣.Proof. Fix a list M ∈ Mµ;µ

π,σ of unhandled and unicellular maps. For eachconnected component of M we use the aforementioned bijection betweensuch maps and oriented maps with arbitrary face-type given by Dołęga[Doł17c, Corollary 3.10]. We get a µ-list of orientable maps with arbitraryface type.

2.3.4. Measure of non-bipartiteness for matchings

The hypothetical statistic wtλ from the Matchings-Jack Conjecture shouldbe a marker of non-bipartiteness for matchings. Naturally, matchings cor-respond to lists of maps, in particular bipartite matching to lists of orientedmaps.

The naive thought how the statistic wtλ should be defined is to adapt themeasure of non-orientability introduced by La Croix by the correspondencebetween matchings and collections of maps given by Corollary 2.12. Regret-fully, the measure introduced by La Croix is defined for lists of rooted maps,however there is no canonical way to create such a list from an element ofM(Gλ;µπ,σ

).

However, there is one special class of matchings, which may be identi-fied with lists of rooted maps, namely Gλ;λ

π,σ . When the number of faces is

35

equal to the number of connected components, numbering and rooting facesoverlap with numbering and rooting components. For a fixed measure ofnon-orientability η we define

statη : Gλ;λπ,σ −→ [d(π, σ;λ)]δ 7−→ statη (δ) := η (Mδ)

For given partitions λ, π, σ ` n we define the following polynomial

(Gη)λ;λπ,σ :=

∑δ∈Gλ;λ

π,σ

βstatη(δ). (2.6)

Definition 2.28. We say that a matching δ ∈ Gλ;λπ,σ is unhandled if the corre-

sponding map Mδ ∈Mλ;λπ,σ is so.

Lemma 2.29. For any triple of partitions π, σ, λ ` n the correspondingpolynomial (Gη)λ;λ

π,σ is of degree at most d(π, σ;λ). Moreover, the matchingδ contributes to the ground term if and only if δ is bipartite. The match-ing δ contributes to the leading coefficient if and only if δ is an unhandledmatching.

Moreover, the top-degree coefficient may be enumerated in two differentmanners:∣∣∣δ ∈ Gλ;λ

π,σ : δ is unhandled∣∣∣ =

∑ν:ν�λ

zλzν

∣∣∣δ ∈ Gν;λπ,σ : δ is bipartite

∣∣∣.Proof. Observe that for fixed measure of non-orientability η polynomials(Gη)λ;λ

π,σ and (Hη)λ;λπ,σ are equal. The first statement follows immediately from

Lemma 2.26. The second statement is an easy conclusion of Corollary 2.27and relation given in Observation 2.17.

36

Chapter 3

Structure constants of Jackcharacters

3.1. Jack characters and structure constantsIn this section we present the notion of Jack characters and their structureconstants.

3.1.1. Jack characters

We expand Jack polynomial in the basis of power-sum symmetric functions:

J(α)λ =

∑µ

θ(α)µ (λ) pµ. (3.1)

The above sum runs over partitions µ such that |µ| = |λ|. The coefficientθ

(α)µ (λ) is called unnormalized Jack character.

Jack characters θ(α)µ provide a kind of dual information about the Jack

polynomials. Better understanding of the combinatorics of Jack charactersmay lead to a better understanding of Jack polynomials themselves. Thiskind of approach may be traced back to the work of Kerov and Olshanski[KO94]. For a fixed conjugacy class µ they considered characters of thesymmetric group evaluated on µ. This is opposite to the usual way of viewingthe characters of the symmetric groups, namely to fix the representation λand to consider the character as a function of the conjugacy class µ. Lassalle[Las08, Las09] adapted idea of Kerov and Olshanski to the framework of Jackcharacters.

37

As Jack symmetric functions(J

(α)λ

)λform a basis of the symmetric

functions, the functions(θ

(α)µ

)µ`n

form a basis of the algebra of functionson Young diagrams with n boxes [Fér12, Proposition 4.1]. Dołęga and Féray[DF16, Appendix B.2] showed that the coefficients appearing in the expan-sion of a pointwise product of two unnormalized Jack characters in theunnormalized Jack character basis coincide with the connection coefficientsfrom (2.1), namely

θ(α)π · θ(α)

σ =∑µ`n

cµπ,σθ(α)µ .

for all triples of partitions π, σ, µ ` n. This observation encourages us tolook more closely into the field of connection coefficients via the context ofJack characters.

3.1.2. Normalized Jack characters

We define Jack characters Chπ by a choice of the normalization of θ(α)π . We

will use the normalization introduced by Dołęga and Féray [DF16] whichoffers some advantages over the original normalization of Lassalle. There-fore, with the right choice of the multiplicative constant, the unnormalizedJack character θ(α)

λ (π) from (3.1) becomes the normalized Jack characterCh(α)

π (λ), defined as follows.Definition 3.1. For a given number α > 0 and a partition π, the normalizedJack character Ch(α)

π (λ) is defined by:

Ch(α)π (λ) :=

1√α

|π|+`(π)(|λ| − |π|+m1(π)

m1(π)

)zπ θ

(α)π∪1|λ|−|π|(λ) if |λ| ≥ |π|,

0 if |λ| < |π|,

where zπ is the standard numerical factor, and ∪ denotes concatenation oftwo partitions, see Section 2.1.1. The choice of an empty partition π = ∅ isacceptable; in this case Ch(α)

∅ (λ) = 1.

3.1.3. The deformation parameters

In order to avoid dealing with the square root of the variable α, we introducean indeterminate A such that A2 := α. Jack characters are usually defined interms of the deformation parameter α. After the substitution α := A2, eachJack character becomes a function of A. In order to keep the notation light,we will make this dependence implicit and we will simply write Chπ(λ).

38

The algebra of Laurent polynomials in the indeterminate A will be de-noted by Q

[A,A−1]. For an integer d we will say that a Laurent polynomial

f =∑k∈Z

fkAk ∈ Q

[A,A−1

]is of degree at most d if fk = 0 holds for each integer k > d.

The quantityγ := −A+ 1

A∈ Q

[A,A−1

]and its opposite

δ := A− 1A∈ Q

[A,A−1

].

will play a special role in our setting.

3.1.4. Structure constants

Structure constants gµπ,σ of Jack characters are defined by expansion of thepointwise product of two Jack characters in the basis of Jack characters:

Chπ ·Chσ =∑µ

gµπ,σ(δ) Chµ .

Explicit motivation for studying such quantities comes from a specialchoice of the deformation parameter α = 1, when Jack polynomials coincidewith Schur polynomials. In this case, Frobenius duality ensures that thestructure constants coincide with the connection coefficients for the sym-metric groups [IK99].

Dołęga and Féray proved [DF16, Theorem 1.4] that each structure con-stant gµπ,σ is a polynomial in the variable δ :=

√α− 1√

αof degree bounded

as follows:degδ gµπ,σ ≤ min

i=1,2,3

(ni(π) + ni(σ)− ni(µ)

), (3.2)

where

n1(π) = |π|+ `(π),n2(π) = |π| − `(π),n3(π) = |π| − `(π) +m1(π).

39

For example, we have

Ch3 Ch2 =6δCh3 + Ch3,2 +6 Ch2,1 +6 Ch4,

Ch3 Ch3 =(6δ2 + 3) Ch3 +9δCh2,1 +18δCh4 +3 Ch1,1,1 ++ 9 Ch3,1 +9 Ch2,2 +9 Ch5 + Ch3,3 .

The numerical computations, such as the ones above, suggest that the struc-ture constants of Jack characters might have some algebraic and combi-natorial structure, which was proposed in the following conjecture [Śn16,Conjecture 0.1].

Conjecture 3.2 (Structure constants of Jack characters). For any parti-tions π, σ, µ, the corresponding structure constant

gµπ,σ(δ) ∈ Q[δ]

is a polynomial with non-negative integer coefficients.

3.2. The top-degree part of structure constantsand connection coefficientsIn this section we present Theorem 3.3 which gives us a formula for thetop-degree part of structure constants for Jack characters. Furthermore, inSection 3.2.2 we show the relation between the structure constants for Jackcharacters and the connection coefficients for Jack symmetric functions. Wetranslate Theorem 3.3 into the field of connection coefficients, more precisely,we give a proof of Theorem 2.5.

3.2.1. The second result

We present an explicit formula for the top-degree part of structure constantsof Jack characters.

Let us recall that we present an oriented map as a graph on the planewith a fixed cyclic order of outgoing edges together with a choice of the root,see Figure 2.8. By convention we fixed the counter-clockwise orientationaround vertices or, equivalently, the clockwise orientation of the page, seeFigure 2.8. Similarly, we will present a µ-collections of maps.

Let us recall that M•;µπ,σ denotes the set of all µ-lists of bipartite rootedand oriented maps which satisfy

ΛW(M) = π and ΛB(M) = σ,

see Figure 3.1.

40

1

2

1

2

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

2

1

2

1

(a) All lists of maps in the set M•;µπ,σ∪1 for partitions π = (3, 3), σ = (3, 2), andµ = (3, 3). Those lists of maps consist of two connected components which arenumbered by 1 and 2, each has 3 edges. The vertex structure is given by π and σ.

(b) Maps from the set M•;(3)(3),(3). Each of them can

be rooted in the unique way.

(c) The only map from the set M•;(3)(3),(2,1). It could

be rooted in three different ways.

Figure 3.1 – There are twelve lists of maps in a set M•;µπ,σ∪1 for partitionsπ = (3, 3), σ = (3, 2), and µ = (3, 3), see Figure 3.1a. Each of them consistsof a map from M

•;(3)(3),(3) and M

•;(3)(3),(2,1) presented on Figure 3.1b and Figure 3.1c

respectively.

41

Theorem 3.3 (The second result). For any triple of partitions π, σ, µ, thecorresponding polynomial gµπ,σ(δ) achieves one of the upper bounds on thedegree given in (2.3), namely

d (π, σ;λ) :=(|π| − `(π)

)+(|σ| − `(σ)

)−(|µ| − `(µ)

)if and only if |µ| ≥ |π|, |σ|, and both partitions π ∪ 1|µ|−|π| and σ ∪ 1|µ|−|σ|are sub-partitions of µ, see Definition 2.3. For such partitions, the leadingcoefficient of gµπ,σ(δ) is a positive integer expressed in the following way:[

δd(π,σ;µ)]gµπ,σ = C(π, σ;µ) · zπzσ

zµ

∣∣∣M•;µπ∪1|µ|−|π|,σ∪1|µ|−|σ|

∣∣∣,where

C(π, σ;µ) =m1(µ)∑k=0

(m1(µ)k

)(m1(π) + |µ| − |π| −m1(µ)

m1(π)− k

)·(

m1(σ) + |µ| − |σ| −m1(µ) + k

m1(σ)−m1(π) + k

),

which is equal to (m1(π) + |µ| − |π|

m1(π)

)(m1(σ) + |µ| − |σ|

m1(σ)

)

if m1(µ) = 0 and is equal to 1 if π, σ, µ are partitions of the same integer.

Section 3.3 is devoted to the proof of above theorem.Example 3.4. Let us consider three partitions π = (3, 2), σ = (3, 3), andµ = (3, 3). In Figure 3.1 we have shown that M•;µπ,σ∪1 = 12. Using thetheorem above, the d (π, σ;µ)-coefficient is equal to

[δd(π,σ;µ)

]gµπ,σ = 6 · 18

18

(10

)(00

)12 = 72.

3.2.2. Relations between the structure constants gµπ,σ and the con-nection coefficients cµπ,σIt is worth mentioning that the coefficients cµπ,σ are indexed by three par-titions of the same size, while the quantities gµπ,σ are indexed by triples ofarbitrary partitions. Dołęga and Féray investigated the relationship between

42

these two families of coefficients and showed [DF16, Equation (19)] that forµ, π, σ ` n,

cµπ,σ =√αd(π,σ;µ) zµ

zπzσ

m1(π)∑i=0

gµ∪1iπ,σ · i!

(n− |µ|

i

), (3.3)

where π is constructed from the partition π by deleting all units.Dołęga and Féray [DF16] proven the polynomiality and the bound on the

degree of gµπ,σ. Using (3.3) they deduced the polynomiality and the boundof the degree of connection coefficients cµπ,σ. We establish other relationsbetween those two families of coefficients.

Corollary 3.5. For three given partitions µ, ν, λ ` n, each of the polyno-mials cλµ,ν(β) and gµπ,σ(δ) is of degree at most d(π, σ;µ), and their leadingcoefficients coincide up to a normalizing constant, namely[

βd(π,σ;µ)]cλµ,ν = zµ

zπzσ·[δd(π,σ;µ)

]gµπ,σ.

Proof. Fix three partitions µ, ν, λ ` n. Observe that for each i ≥ 0, thethird estimation shown in (3.2) gives us

degδ gµ∪1iπ,σ ≤ d (π, σ;µ)− i.

Let us recall that δ =√α − 1√

α, hence the right-hand side of (3.3) is

of α-degree at most equal to 2d(π, σ;µ), and in the sum over i, the onlycontribution to the 2d(π, σ;µ)-degree coefficient comes from gµπ,σ. We have

[√α

2d(π,σ;µ)](√αd(π,σ;µ) zµ

zπzσ

m1(π)∑i=0

gµ∪1iπ,σ · i!

(n− |µ|

i

))=

zµzπzσ

[δd(π,σ;µ)

]gµπ,σ.

Since β = α − 1, the 2d(π, σ;µ)-degree coefficient of cλµ,ν in variable√α

coincides with d(π, σ;µ)-degree coefficient in variable β. Hence (3.3) finishesthe proof.

Assuming Theorem 3.3 we are ready to prove the main result of theprevious chapter. The proof may seem intricate, it combines different factswhich have been proven so far.

43

Proof of Theorem 2.5. Fix partitions π, σ, λ ` n. We investigate the poly-nomial cλπ,σ(β). By Corollary 3.5 we have[

βd(π,σ;λ)]cλπ,σ =

zλzπzσ

·[δd(π,σ;λ)

]gλπ,σ,

and by Theorem 3.3 we know that the polynomial gλπ,σ achieves the d(π, σ;λ)-degree part if and only if |λ| ≥ |π|, |σ|, π � λ, and σ � λ. Observe thatthis condition is equivalent to π � λ and σ � λ. Hence the condition onpartitions π, σ, λ for achieving by cλπ,σ the d(π, σ;λ)-degree.

Thus, we have[βd(π,σ;λ)

]cλπ,σ

Corollary 3.5=zλzπzσ

·[δd(π,σ;λ)

]gλπ,σ

Theorem 3.3=∣∣∣M•;λπ,σ∣∣∣.

Since there is only one map M1 ∈ M (1)(1),(1), we have

∣∣∣M•;λπ,σ∣∣∣ =∣∣∣M•;λπ,σ∣∣∣.

Indeed, from any λ-list of mapsM ∈ M•;λπ,σ we can canonically create a λ-listof map M ∈ M•;λπ,σ by erasing the last |λ| − |λ| components. This procedureis reversible, since we can add new M1 components to M . Then we have∣∣∣M•;λπ,σ∣∣∣ =

∑ν:ν�λ

∣∣∣Mν;λπ,σ

∣∣∣ Observation 2.17= zλzν

∑ν:ν�λ

∣∣∣Gν;λπ,σ

∣∣∣.Hence [

βd(π,σ;λ)]cλπ,σ = zλ

zν

∑ν:ν�λ

∣∣∣Gν;λπ,σ

∣∣∣.From Lemma 2.29 we conclude that the leading coefficient of cλπ,σ overlaps

with the leading coefficient of the polynomial

(Gη)λ;λπ,σ :=

∑δ∈Gλ;λ

π,σ

βstatη(δ),

see (2.6), and that both are of the same degree. From Lemma 2.29 wealso get the second expression for the leading coefficient of the polynomialcλπ,σ.

44

3.3. Proof of Theorem 3.3This section is devoted to the proof of Theorem 3.3. Firstly, we present somebasic computations leading to the exact formulas for the top-degree part ofJack characters. We present those formulas in terms of injective embeddingsinto Young diagrams. Secondly, we consider a particular class of collectionsof bipartite maps Pµπ,σ which constitute a good candidate for the top-degreeparts of the structure constants gµπ,σ. Finally, we prove that those candidatesfor the top-degree part of structure constants gµπ,σ (see Proposition 3.16) areindeed them.

3.3.1. Embeddings of bicolored graphs

A bicolored graph G is a bipartite graph together with a choice of the colour-ing of its vertex set V; we denote by V• and V◦ respectively the sets of blackand white vertices of G.Definition 3.6. An injective embedding F of a bicolored graph G to a Youngdiagram λ is a function which maps V◦ to the set of columns of λ, maps V•to the set of rows of λ, and maps injectively the set of edges E to the set ofboxes of λ, see Figure 3.2. We also require that F preserves the relation ofincidence, i.e. each vertex v ∈ V should be mapped to a row or a columnF (v) which contains the box F (e), for every edge e ∈ E incident to v. Wedenote by NG(λ) the number of such embeddings of G into λ.

It is also useful to consider injective embeddings of a graph G into aYoung diagram λ, with the roles of black and white vertices reversed (i.e.black vertices are mapped into columns, white vertices into the rows). Werefer to such embeddings as negative injective embeddings and denote thenumber of such embeddings as NG(λ).Definition 3.7. For any partition π = (π1, . . . , πr) we define the graph Gπas the unique bicoloured graph consisting of r black vertices of degreesπ1, . . . , πr respectively and |π| white vertices, each of degree one (see Fig-ure 3.2). Similarly, we define Gπ as the unique bicoloured graph consistingof r white vertices of degrees π1, . . . , πr respectively and |π| black vertices,each of degree one.Remark 3.8. The number NGπ (λ) of injective embeddings of the graph Gπinto the Young diagram λ is equal to the number NGπ

(λ) of negative injec-tive embeddings of the graph Gπ into the Young diagram λ.

45

e1

e2e3

e4v•2

v•1

v◦4

v◦3

v◦2

v◦1

e3 e2 e1

e4

v◦3,v◦4 v◦2 v◦1

v•1

v•2

Figure 3.2 – The graph Gπ associated with the partition π = (3, 1). Onthe right, an example of its injective embedding into the Young diagramλ = (4, 3).

3.3.2. Exact formulas for top-degree part of Jack characters

Śniady proved [Śn15, Proposition 3.5] that each Jack character is a functionon the set Y of Young diagrams

Y 3 λ 7−→ Chπ(λ) ∈ Q[A,A−1

]|π|−`(π)

with values in the set Q[A,A−1]

|π|−`(π) of Laurent polynomials in the vari-able A of degree at most |π| − `(π). We denote by[

Atop]

Chπ (λ) :=[A|π|−`(π)

]Chπ (λ)

the leading part of this Laurent polynomial. We shall express this quantityin terms of injective embeddings of Gπ into λ.

Proposition 3.9. For any Young diagram λ ∈ Y and partition π, we havethat [

Atop]

Chπ (λ) = NGπ (λ) .

That is, the leading part of Chπ(λ) is equal to the number of injective em-beddings of the graph Gπ into the Young diagram λ.

Example 3.10. Let us consider the partition π = (3, 1) and the Young dia-gram λ = (λ1, λ2). We have[

Atop]

Ch(3,1) (λ1, λ2) = NGπ (λ) = λ14 + λ1

3 · λ21

+ λ11 · λ2

3 + λ24.

One of embeddings which contributes to NGπ(λ) is presented on Figure 3.2.

46

Before proving Proposition 3.9 we introduce the notion of α-shifted sym-metric functions (see more in [Las08, Section 2.2] or [AF17, Definition 2.2])and present Jack characters in this context.Definition 3.11. An α-shifted symmetric function F = (FN )N≥1 is a se-quence of polynomials FN such that

• for each N ≥ 1, FN is a polynomial in N variables x1, . . . , xN with co-efficients in the field of rational functions Q(α) in some indeterminateα that is symmetric in the variables

ξ1 := x1 −1α, ξ2 := x2 −

2α, . . . , ξN := xN −

N

α,

• for each N ≥ 1, FN+1(x1, . . . , xN , 0) = FN (x1, . . . , xN ) (the stabilityproperty),

• supN≥1 deg(FN ) <∞.

The degree of a shifted-symmetric function F is defined as maximum of thedegrees of the corresponding polynomials FN (x1, . . . , xN ).

Śniady and Féray gave some abstract characterizations of Jack characters[Śn15, Theorem 1.7, Theorem A.2]. We present the one given by Féray,which can be traced back to the earlier work of Knop and Sahi [KS96].

Theorem 3.12. [Śn15, Theorem A.2] Let π be a partition and A be acomplex number such that − 1

α= − 1

A2 is not a positive integer. Thereexists a unique shifted-symmetric function F such that:

(J1) F is a shifted-symmetric function of degree |π|, and its top-degree ho-mogeneous part is equal to

A|π|−`(π) pπ(λ1, . . . , λm),

where pπ is the power-sum symmetric polynomial given by the formula

pπ (λ) =∏r

∑i

λπri .

(J2) F (λ) = 0 holds for each Young diagram λ such that |λ| < |π| (thevanishing property).

Moreover, if α is a positive real number, the function F = (FN )N≥1satisfies Chπ(λ) = Fr(λ1, . . . , λr) for each Young diagram λ = (λ1, . . . , λr).

47

To keep notation short, we introduce the following symmetric function

pπ (λ) :=∗r

∑i

λπri ,

where

λlii ∗ λ

lj

j =

λli+lji if i = j,

λlii · λ

lj

j otherwise,and

λl = λ · (λ− 1) · · · (λ− l + 1)︸ ︷︷ ︸l factors

.

Proof of Proposition 3.9. Observe that

pπ (λ) = NGπ (λ) .

We will show that [Atop

]Chπ (λ) = pπ (λ) .

Let F be an α-shifted symmetric function associated to π by Defini-tion 3.11. Let us choose a sufficiently large integer N , e.g. N > |π|. Let ustreat the coefficients of the polynomial FN as variables. The equality

FN (λ) = Chπ(λ) ∈ Q[A,A−1

]|π|−`(π)

,

which holds for each λ ∈ Y, becomes a system of equations with coefficientsin N≥0. This system is large enough to conclude that each coefficient of apolynomial FN is a linear combination of the quantities Chπ(λ) overQ, henceFN is a polynomial in N variables with coefficients in Q

[A,A−1]

|π|−`(π).Notice that formally we have equality for all α > 0. However, the rational

function from Q(α) is uniquely determined by its values for α ≥ 0.Since FN is a shifted-symmetric function with coefficients in the set

Q[A,A−1]

|π|−`(π), its A-top degree[Atop

]FN (λ1, . . . , λN ) :=

[A|π|−`(π)

]FN (λ1, . . . , λN )

is a symmetric function in the variables λ1, . . . , λN . Indeed, for each per-mutation σ of [N ] we have[

Atop]FN (x1, . . . , xN ) =

[Atop

]FN

(x1 −

1A2 , . . . , xN −

N

A2

)=[

Atop]FN

(xσ(1) −

σ(1)A2 , . . . , xσ(N) −

σ(N)A2

)=[

Atop]FN

(xσ(1), . . . , xσ(N)

).

48

Since FN is of a degree |π|, the polynomial[Atop]FN has the same bound

of the degree. Observe that the homogeneous top-degree part of pπ (λ) isequal to pπ (λ) and so does the homogeneous top-degree part of

[Atop]FN .

Polynomials pπ (λ) and[Atop]FN are both symmetric, hence[

Atop]FN − pπ

is a symmetric polynomial in variables (λ1, . . . , λN ) of a degree at most|π| − 1.

We use the following notation:

Y0 :={

(λ1, . . . , λN ) ∈ ZN : λ1 ≥ . . . ≥ λN ≥ 0 and λ1 + . . .+ λN < |π|}.

By the vanishing property we have[Atop

]FN (λ) =

[Atop

]Chπ (λ) = 0

for all elements λ ∈ Y0. Since there are no injective embeddings of Gπ into aYoung diagram with the number of boxes smaller then the number of edgesin Gπ, we have

pπ (λ) = NGπ (λ) = 0

for all elements λ ∈ Y0. From that we deduce that Y0 is a set of zeros ofthe polynomial

[Atop]FN − pπ. The appropriate set of zeros of a polynomial

of sufficiently small degree determines the vanishing of the polynomial. Infact, we can use the characterisation given by Śniady [Śn15, Lemma 7.1] toconclude that the symmetric polynomial

W (x1, . . . , xN ) :=( [Atop

]FN − pπ

)(x1 − 1, . . . , xN − 1),

which is of a degree at most |π| − 1, vanishes. Hence we conclude that[Atop

]FN = pπ

which finishes the proof.

3.3.3. Hands-shaking procedure

Letπ = (π1, . . . , πn) , σ = (σ1, . . . , σl)

be two partitions. We define a class of collections of maps by the followingprocedure:

49

1. For each i ∈ [n] we assign a white vertex with πi outgoing half-edges.We label this vertex by the number i and we root it, i.e. we choose oneof the outgoing half-edges and decorate it. Similarly, for each j ∈ [l]we assign a black vertex with σj outgoing half-edges and we root it.

2. We match some of the half-edges going out from the white verticeswith some of those going out from black vertices.

3. We close each of the non-closed half-edges by a white or a black vertexso that the graph remains bipartite.

We call the procedure described above the “hands-shaking procedure”.The name provenance could be explained as follows: there are white andblack vertices with hands; the number of hands is given by the partitions πand σ. They shake theirs hands in any way they like, but only black-whiteconnections are allowed. On Figure 3.3 we present an example of applyingthis procedure.Definition 3.13. For a given triple of partitions π, σ, µ we denote by Pµπ,σthe set of all µ-collections of maps which may be obtain as an outcome ofperforming the above presented “hands-shaking procedure”.

Each µ-collection of maps M ∈ Pµπ,σ can be obtained in the unique wayas an outcome of the presented procedure. The uniqueness follows from thefact that the position of each edge from M is uniquely determined by thelabellings on the rooted vertices and the order of the outgoing half-edges.

Observation 3.14. For given partitions π, σ, µ the set Pµπ,σ is non-empty ifand only if the following conditions holds

1. |π|, |σ| ≤ |µ|,

2. both partitions π ∪ 1|µ|−|π| and σ ∪ 1|µ|−|σ| are sub-partitions of µ.

Proof. Firstly, we will show the necessity of conditions. Observe that byperforming “hands-shaking procedure”, in which we obtain a µ-collection ofmap, the vertex set is given by

ΛW(M) = π ∪ 1|µ|−|π| and ΛB(M) = σ ∪ 1|µ|−|σ|.

The first condition follows immediately. Partitions describing white or blackvertices distributions are sub-partitions of a partition describing face distri-bution. Hence the second condition has to be satisfied.

For partitions satisfying those two conditions one can exhibit a collectionof maps from Pµπ,σ, which proves the sufficiency of those conditions.

50

1

2

32

11

2

32

1

1

2

32

1

2 23

1 1

Step 1. Black and whitevertices with outgoinghalf-edges of degrees (σi)and (πj) respectively.

Step 2. Some of the out-going half-edges werematched. The crossingof edges is not important.

Step 3. The rest of out-going half-edges is closed.The collection of maps isbipartite.

As an outcome we obtainthe following collection oftwo maps drawn on a pair ofspheres.

Figure 3.3 – The three steps of “hands-shaking procedure”. As an outputwe obtain the (4, 2)-collection of bipartite maps. Vertices are labelled androoted as as the “hands-shaking procedure” describes.

51

Observation 3.15. For given partitions π, σ, µ: |π|, |σ| ≤ |µ| we have∣∣∣Pµπ,σ∣∣∣ = C(π, σ;µ) · zπzσzµ

∣∣∣M•;µπ∪1|µ|−|π|,σ∪1|µ|−|σ|

∣∣∣where

C(π, σ;µ) =m1(µ)∑k=0

(m1(µ)k

)(m1(π) + |µ| − |π| −m1(µ)

m1(π)− k

)·(

m1(σ) + |µ| − |σ| −m1(µ) + k

m1(σ)−m1(π) + k

), (3.4)

which is equal to (m1(π) + |µ| − |π|

m1(π)

)(m1(σ) + |µ| − |σ|

m1(σ)

)

if m1(µ) = 0 and is equal to 1 if π, σ, µ are partitions of the same integer.

Proof. Observe that the elements of Pµπ,σ are µ-collection of bipartite ori-entable maps whose vertex set is given by

ΛW(M) = π ∪ 1|µ|−|π| and ΛB(M) = σ ∪ 1|µ|−|σ|.

Each such an element has the following labels and roots on the vertices andhalf-edges:

1. there are n white vertices of degrees π1, . . . , πn, each being labelled bya relevant natural number from [n] and rooted, i.e. we choose one ofthe outgoing half-edges and decorate it by an arrow,

2. there are l black vertices of degrees σ1, . . . , σl, each being labelled bya relevant natural number from [l] and rooted.

Moreover, each connected component of an element from Pµπ,σ has at leastone decorated vertex.

We use the double counting method as in Observation 2.15. For eachM ∈ Pµπ,σ we can root and number the connected components in zµ ways.

Let us choose M ∈ M•;µπ∪1|µ|−|π|,σ∪1|µ|−|σ| . The procedure of labelling and

rooting the vertices is much more subtle. Firstly, we have to choose m1(π)

52

fM(e1)

fM (e2)

fM (e3 )

fM (e4)

e1

e2

e4

e3

1

2

2

1

Figure 3.4 – Example of a collection of mapsM ∈ P (3,1)(2,1),(2,1) and an example

of a bijection fµM between the edges in M and the edges of the graph G(3,1).

white (respectively m1(σ) black) vertices and label them by adequate num-bers. At the first sight, we could do this in(

m1(π) + |µ| − |π|m1(π)

)(m1(σ) + |µ| − |σ|

m1(σ)

)zπzσ

ways (which is equal to zπzσ if π, σ, µ are partitions of the same integer).However, in the definition of Pµπ,σ we required to contain at least one labelledvertex from each connected component. This is trivially satisfied if m1(µ) =0. This consideration yields the expression (3.4). We describe briefly thedetails.

There is m1(µ) one-element connected components in M . Denote theset of those components by M1. For each integer k: 0 ≤ k ≤ m1(µ) wecan choose k white vertices from M1 and we required that exactly thosewhite vertices among all white vertices inM1 are numbered. The number ofpossible ways of numbering vertices of M in such a way is the contributionto the sum in (3.4) relevant to k. We sum up over all k = 0, . . . ,m1(µ).

3.3.4. Proof of Theorem 3.3

We prove that candidates pµπ,σ := |Pµπ,σ| for top-degree part of structureconstants gµπ,σ suit well for that role.

Proposition 3.16. For any Young diagram λ ∈ Y, the following equalityholds: [

Atop]

Chπ (λ) ·[Atop

]Chσ (λ) =

∑µ

pµπ,σ

[Atop

]Chµ (λ) . (3.5)

53

Proof. According to Proposition 3.9, the two quantities[Atop

]Chσ (λ) = NGσ (λ) and

[Atop

]Chµ (λ) = NGµ (λ)

can be represented equivalently by the number of injective embeddings ofGσ and Gµ into λ. Similarly,[

Atop]

Chπ (λ) = NGπ(λ)

is equal to the number of negative injective embeddings of Gπ into λ (seeRemark 3.8).

For each M ∈ Pµπ,σ we choose some bijection fµM between the edges ofM and the edges of the graph Gµ (see Definition 3.7), which preserves theconnected components, see Figure 3.4.

We shall construct a bijection between:

• a pair(NGπ

(λ), NGσ (λ))consisting of negative injective embeddings

and injective embeddings of Gσ and Gπ into λ respectively;

• a pair(Pµπ,σ, NGµ (λ)

)consisting of collections of maps from the class

Pµπ,σ and injective embeddings of Gµ into λ.

Construction of such bijection follows the statement of Proposition 3.16.We proceed analogously as in the “hands-shaking procedure” described inSection 3.3.3.

For each i ∈ [n] we assign a white vertex with πi outgoing half-edges. Welabel this vertex by a number i and root it, i.e. we choose one of outgoinghalf-edges and label it. We can choose a bijection between such half-edgesand the edges in Gπ which preserves the connected components. Similarly,for each j ∈ [l] we assign a black vertex with σj outgoing half-edges and weroot it. Then we choose a bijection between such half-edges and the edgesin Gσ which preserves the connected components.

A reverse injective embedding of Gπ and an injective embedding of Gσinto λ transfer into an injective embedding of above described half-edgesgoing out from labelled and rooted black and white vertices.

We use the procedure described in Section 3.3.3 to connect in the uniqueway those outgoing half-edges which are embedded in the same box of Youngdiagram λ. We close each of non-closed half-edges by a white or a blackvertex so that the graph remains bipartite.

In that way we obtain a list of maps M ∈ Pµπ,σ injectively embeddedinto the Young diagram λ. Observe that all edges from any given connected

54

component of M are embedded into the boxes of λ which are in the samerow. Using the bijection fµM between the edges of M and the edges of Gµ,we obtain the injective embedding of Gµ into λ.

The above procedure is reversible. Indeed, for a given collection of mapsM ∈ Pµπ,σ and an injective embedding of Gµ into diagram λ, we can easilyconstruct the injective embedding of the edges of M into the diagram λ, forwhich all edges from any given connected component of M are embeddedto the boxes from the same row. From such an object we can recover theelements from NGσ (λ) and NGπ

(λ).

With Proposition 3.16 in hand, we are ready to present the proof ofTheorem 3.3.

Proof of Theorem 3.3. The upper bound of a degree for polynomials gµπ,σ(δ)

is given in (3.2). Since δ = 1A−A, we have the following estimation

degA gµπ,σ = degδ gµπ,σ ≤ d (π, σ;µ) .

Let us fix a Young diagram λ. Recall that the evaluation of Chπ on anyYoung diagram λ is a Laurent polynomial in Q

[A,A−1] of a degree at most

n2(π) := |π| − `(π). We investigate the n2(π) + n2(σ) degree part of thepointwise product of two Jack characters, namely[

An2(π)+n2(σ)]

Chπ (λ) · Chσ (λ) =[An2(π)+n2(σ)

]∑µ

gµπ,σ Chµ (λ) .

By the estimations on the upper bounds of the A-degrees of Laurent poly-nomials Chπ(λ) and gµπ,σ we have[

Atop]

Chπ (λ) ·[Atop

]Chσ (λ) =

∑µ

[Ad(π,σ;µ)

]gµπ,σ

[Atop

]Chµ (λ) .

We compare the above equation with Proposition 3.16 and we get∑µ

[Ad(π,σ;µ)

]gµπ,σ Chµ (λ) =

∑µ

pµπ,σ Chµ (λ) .

Recall that Chµ(λ) = pµ(λ). We have∑µ

[Ad(π,σ;µ)

]gµπ,σpµ (λ) =

∑µ

pµπ,σpµ (λ) , (3.6)

55

The function pµ (λ) is symmetric and its homogeneous top-degree partcoincides with the power-sum symmetric polynomial pµ. This coincidencetogether with the fact that power-sum symmetric functions form a basis ofsymmetric functions allows us to deduce that functions pµ (λ) form also sucha basis. We may look at (3.6) as on the equality of symmetric functions.Since the basis determines its coefficients in the unique way, we concludethat [

Ad(π,σ;µ)]gµπ,σ = pµπ,σ.

The d(π, σ;µ)-degree coefficients in variable A and δ of gµπ,σ are equal. Weconclude

δ(π,σ;µ)gµπ,σ = pµπ,σ.

Observation 3.15 and Observation 3.14 finish the proof.

56

Chapter 4

Algebras with twomultiplications and theircumulants

4.1. Algebras with two multiplicationsIn this section we shall present the notion of cumulants in probability the-ory. In Section 4.1.3 we define algebras with two multiplications and theircumulants. In Theorem 4.6 we present some cumulant formula which holdsin algebras with two multiplications.

4.1.1. Cumulants in probability theory

One of classical problems in probability theory is to describe the joint dis-tribution of a family (Xi) of random variables in the most convenient way.Common solution of this problem is to use the family of moments, i.e. theexpected values of products of the form

E (Xi1 · · ·Xil) .

It has been observed that in many problems it is more convenient to make useof the cumulants [Hal81, Fis28], defined as the coefficients of the expansionof the logarithm of the multidimensional Laplace transform around zero:

κ (X1, . . . , Xn) := [t1 · · · tn] logEet1X1+···+tnXn

= ∂n

∂t1 · · · ∂tnlogEet1X1+···+tnXn

∣∣∣∣∣t1=...=tn=0

,(4.1)

57

where the terms on the right-hand side should be understood as a formalpower series in the variables t1, . . . , tn. Cumulant is a linear map with respectto each of its arguments.

There are some good reasons for claiming advantage of cumulants overthe moments. One of them is that the convolution of measures correspondsto the product of the Laplace transforms or, in other words, to the sum of thelogarithms of the Laplace transforms. It follows that the cumulants behavein a very simple way with respect to the convolution, namely cumulantslinearize the convolution.

Cumulants allow also a combinatorial description. One can show thatthe expression (4.1) is equivalent to the following system of equations, calledthe moment-cumulant formula:

E (X1 · · ·Xn) =∑ν

∏b∈ν

κ (Xi : i ∈ b) (4.2)

which should hold for any choice of the random variables X1, . . . , Xn whosemoments are all finite. The above sum runs over the set partitions ν of theset [n] = {1, . . . , n} and the product runs over the blocks of the partition ν.Example 4.1. For three random variables the corresponding moment expandsas follows:

E (X1X2X3) = κ(X1 ) · κ(X2 ) · κ(X3 ) + κ(X1, X2 ) · κ(X3 )+κ(X2, X3 ) · κ(X1 ) + κ(X1, X3 ) · κ(X2 )+κ(X1, X2, X3 ).

The moment-cumulant formula defines the cumulant κ(X1, . . . , Xn ) induc-tively according to the number of arguments n.

4.1.2. Conditional cumulants

Let A and B be commutative unital algebras and let E : A −→ B be a unitallinear map. We say that E is a conditional expected value. For any tuplex1, . . . , xn ∈ A we define their conditional cumulant as

κ (x1, . . . , xn) = [t1 · · · tn] logEet1x1+···+tnxn

= ∂n

∂t1 · · · ∂tnlogEet1x1+···+tnxn

∣∣∣∣∣t1=...=tn=0

∈ B (4.3)

where the terms on the right-hand side should be understood as in Eq. (4.1).In this general approach, cumulants give a way of measuring the discrepancybetween the algebraic structures of A and B.

58

4.1.3. Framework

We are interested in a following particular case. We assume that A is a lin-ear space equipped with two commutative multiplication structures, whichcorrespond to two products: · and ∗. Together with each multiplication Aform the commutative algebra. We call such structure an algebra with twomultiplications. We also assume that the mapping E is the identity map onA:

E : (A, ·) id−→ (A, ∗) .

In this case the cumulants measure the discrepancy between these two mul-tiplication structures on A. This situation arises naturally in many branchesof algebraic combinatorics, for example in the case of Macdonald cumulants[Doł17a, Doł17b] and cumulants of Jack characters [DF17, Śn16].

Since the mapping E is the identity, we can define cumulants of cumu-lants and further compositions of them. The terminology of cumulants ofcumulants was introduced in [Spe83] and further developed in [Leh13] (calledthere nested cumulants) in a slightly different situation of an inclusion of al-gebras C ⊆ B ⊆ A and conditional expectations A E1−→ B E2−→ C.

As we already mentioned in Section 4.1.1, cumulants allow also a com-binatorial description via the moment-cumulant formula. When E is theidentity map (4.3) is equivalent to the following system of equations:

a1 ∗ · · · ∗ an =∑ν

∏b∈ν

κ (ai : i ∈ b) , (4.4)

for any ai ∈ A (the product on the right-hand side is the ·-product). Theabove sum runs over the set partitions ν of the set [n] and the product runsover the blocks of the partition ν.

Let A be a multiset consisting of elements of the algebra A. To simplifynotation, for any partition ν of a multiset A we introduce the correspondingcumulant κν as the product:

κν =∏b∈ν

κ (a : a ∈ b) .

We denote by P (A) the set of all partitions of A. With this notation, themoment-cumulant formula has the following form:

∗a∈A

a =∑

ν∈P(A)κν . (4.5)

59

Example 4.2. Given three elements a1, a2, a3 ∈ A, we have:

a1 ∗ a2 ∗ a3 = κ(a1 ) · κ(a2 ) · κ(a3 ) + κ(a1, a2 ) · κ(a3 ) +κ(a2, a3 ) · κ(a1 ) + κ(a1, a3 ) · κ(a2 ) +κ(a1, a2, a3 ).

4.1.4. The third result

We present an algebraic formula which involves two multiplications on linearspace A: (

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

),

as a sum of products of only one type of multiplication.We use the following notation. We denote by A1, . . . , An multisets con-

sisting of elements of A. We denote by A = A1 ∪ · · · ∪ An the multiset,corresponding to the sum of all multisets Ai. We use also the followingnotation for elements of Ai:

Ai ={ai1, . . . , a

iki

},

hence the multiset A consists of the following elements:

A ={a1

1, . . . , a1k1 , . . . , a

n1 , . . . , a

nkn

}.

Due to a combinatorial nature of this result we introduce now the defi-nitions of the mixing reduced forests and theirs cumulants. We begin withthe following definition.Definition 4.3. Consider a forest F whose leaves are labelled by elements ofan algebra A. We denote by A the multiset consisting of labels of all leaves.If each node (vertex which is not a leaf) of F , has at least two descendants,we call F a reduced forest with leaves in A. We denote the set of such forestsby F (A).

For a reduced forest F ∈ F (A) we associate a cumulant κF in thefollowing way:Definition 4.4. Consider a reduced forest F ∈ F (A). Denote by av the labelof a leaf v. For any vertex v ∈ F we define inductively the quantities κv asfollows:

κv :={av if v is a leaf,κ(κv1 , . . . , κvn ) otherwise,

60

where v1, . . . , vn are the descendants of v. For the whole forest F , we definethe cumulant κF to be the product:

κF := ∗iκVi ,

where Vi are the roots of all trees in F .Finally, we introduce a class of the mixing forests and the associated

quantity wF .Definition 4.5. Consider a multiset A = A1 ∪ · · · ∪An and a reduced forestF ∈ F (A). We say that F is mixing for a division A1, . . . , An (or shortlymixing) if for each vertex v whose descendants are all leaves, those descen-dants are elements of at least two distinct multisets Ai and Aj . Denote byF(A) the set of all reduced mixing forests.

For a reduced mixing forest F we define the quantity wF to be thenumber of vertices in F minus the number of leaves (see Figure 4.1).

We are ready to formulate the main result of this chapter.

Theorem 4.6 (The third result). Let A1, . . . , An be multisets consisting ofelements of A. Let A be the sum of those multisets. Then:(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑F∈F(A)

(−1)wF κF .

Example 4.7. Figure 4.1 presents all reduced forests F on the multiset A ={a1

1, a12, a

21}. Six of them are mixing. By the statement of the theorem, we

have (a1

1 ∗ a12)· a2

1 = a11 ∗ a1

2 ∗ a21 − κ(a1

1, a21 ) ∗ κ(a1

2 )−κ(a1

2, a21 ) ∗ κ(a1

1 )− κ(a11, a

12, a

21 )

+κ(κ(a1

1, a22 ), κ(a1

2 ))

+ κ(κ(a1

2, a22 ), κ(a1

1 )).

4.1.5. How to prove Theorem 4.6?

Theorem 4.6 is a straightforward conclusion from two propositions which wepresent in this section. In our opinion they are interesting themselves.

We introduce a gap-free vertex colouring on forests F ∈ F (A).Definition 4.8. For a reduced forest F with leaves in a multiset A = A1 ∪· · · ∪An we say that c is a gap-free vertex colouring with length r if

• c is a coloured by the numbers {0, . . . , r} and each colour is used atleast once;

61

a11 a1

2 a21

κ(κ(a1

1, a12 ), a2

1

)

a11 a1

2 a21

κ(a1

1, κ(a12, a

21 ))

wF = 2

a11 a1

2 a21

κ(a1

2, κ(a11, a

21 ))

wF = 2

a11 a1

2 a21

κ(a11, a

12, a

21 )

wF = 1

a11 a1

2 a21

κ(a11, a

12 ) ∗ a2

1

a11 a1

2 a21

a11 ∗ κ(a1

2, a21 )

wF = 1

a11 a1

2 a21

a12 ∗ κ(a1

1, a21 )

wF = 1

a11 a1

2 a21

a11 ∗ a1

2 ∗ a21

wF = 0Not treesTrees

Figure 4.1 – All reduced forests on A = A1 ∪ A2 = {a11, a

12, a

21}. Six of

them (on the right-hand side) are mixing; we present theirs wF numbers.The remaining two elements (shown on the left-hand side) are not mixing.We also present the corresponding cumulants κF . Observe that, among allreduced forests F ∈ F (A), exactly half, presented on top, consists of a singletree (see Remark 4.39).

62

• each leaf is coloured by 0;

• the colours are strictly increasing on any path from the root to a leaf.

We denote by |c| := r the length of c . We call such a colouring c weakly-mixing if it satisfies one of the following additional conditions:

1. either there exists a vertex coloured by 1 with at least two descendants,each of whom belongs to a distinct multiset Ai,

2. or colouring c does not use the colour 1 at all.

We denote by CF the set of all gap-free and weakly-mixing colourings of aforest F .

The following result is a juggling of a concept of cumulants. We presentits proof in Section 4.4.

Proposition 4.9. Let A1, . . . , An be multisets consisting of elements of A.Let A be a sum of those multisets. Then(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑F∈F(A)

κF∑c∈CF

(−1)|c|. (4.6)

In Section 4.5, we will show that summing over all colourings c ∈ CF fora reduced forest F ∈ F (A) gives a surprisingly simple number. This resultis presented in proposition below.

Proposition 4.10. Let A1, . . . , An be multisets consisting of elements of A.Let A be a sum of those multisets. Then, for any reduced forest F ∈ F (A),the following holds:

∑c∈CF

(−1)|c| ={

(−1)wF if F ∈ F (A) ,0 otherwise.

Observe that combing Proposition 4.9 and Proposition 4.10 we obtainthe statement of Theorem 4.6.

4.2. Analogue of Leonov–Shiryaev’s formulaIn this section we present the well-known cumulant formula given by Leonovand Shiryaev in 1959. Theorem 4.16 is a reformulation of Theorem 4.6 andmay be seen as an analogue of Leonov–Shiryaev’s formula.

63

4.2.1. Leonov–Shiryaev’s formula

In 1959 Leonov and Shiryaev [LS59, Equation IV.d] presented a formula fora cumulant of products of random variables:

κ (X1,1 · · ·Xk1,1, . . . , X1,n · · ·Xkn,n)

in terms of simple cumulants. This formula was first proved by Leonov andShiryaev [LS59], a more direct proof was given by Speed [Spe83]. The tech-nique of Leonov and Shiryaev was used in many situations [SSR88, Leh04]and was further developed in other papers: Krawczyk and Speicher [KS00,MST07] found the free analogue of the formula; the formula was furthergeneralized to the partial cumulants [NS06, Proposition 10.11].

We briefly present the original formula stated by Leonov and Shiryaevin the framework of an algebra with two multiplications. We use the samenotation for multisets A1, . . . , An and its sum A = A1 ∪ · · · ∪ An as inSection 4.1.4.

We introduce a notion of a strongly-mixing partitions (called also inde-composable partitions).Definition 4.11. Consider a multiset A = A1 ∪ · · · ∪An and any partition νof A. A partition λ = {λ1, λ2} is called a row partition if for each multisetAi we have: either Ai ⊆ λ1 or Ai ⊆ λ2.

A partition ν = {ν1, . . . , νq} is called a strongly-mixing partition for thedivision A = A1 ∪ · · · ∪ An (or shortly strongly-mixing partition), if thereis no row partition λ such that for any i either νi ∈ λ1, or νi ∈ λ2 (seeFigure 4.2).

We denote by P (A) the set of all strongly-mixing partitions of a set A.We can now express the Leonov–Shiryaev’s formula in a framework rel-

evant to this dissertation.

Theorem 4.12 (Leonov–Shiryaev’s formula).

κ(a1

1 · · · a1k1 , . . . , a

n1 · · · ankn

)= κ

ki∏j=1

aji : i ∈ [n]

=∑

ν∈P(A)

κν , (4.7)

where the sum on the right-hand side is running over all strongly-mixingpartitions of a set A.

Example 4.13. By Leonov–Shiryaev’s formula the cumulant κ(a1

1 · a12, a

21)

expresses as follows:

κ(a1

1 · a12, a

21)

= +κ(a11, a

22 ) · κ(a1

2 ) + κ(a12, a

22 ) · κ(a1

1 )+κ(a1

1, a12, a

21 ).

64

ν ={{a1

1, a21}, {a3

1, a22, a

32}, {a4

1, a51}, {a5

2}}

λ ={{a1

1, a21, a

22, a

31, a

32}, {a4

1, a51, a

52}}

a41 a5

1

a52

a11 a2

1 a31

a22 a3

2

Figure 4.2 – The multiset A = {a11, a

21, a

22, a

31, a

32, a

41, a

51, a

52} and the set par-

tition ν. There exists a row partition λ (dashed line) such that each partof ν is contained in one of the parts of λ. Hence the partition ν is notstrongly-mixing.

4.2.2. Analogue of Leonov–Shiryaev’s formula

Leonov–Shiryaev’s formula relates a cumulant of products with some prod-ucts of cumulants. In the situation we investigate, where the conditionalexpected value is the identity mapping, we can define two types of cumu-lants. For each of them we have Leonov–Shiryaev’s formula. We presentnow the third formula, which is a mix of those two.

Consider the identity map:

(A, ·) id−→ (A, ∗)

between commutative unital algebras (A, ·) and (A, ∗). Equation (4.4) de-fined cumulants κ of the identity mapping. Observe that we can also considerthe inverse mapping, namely the map:

(A, ∗) id−1−→ (A, ·).

This mapping gives us a way to define cumulants (according to (4.4)), whichwe denote by κ∗.

We present below the Leonov–Shiryaev’s formula for both mappingsmentioned above:

κ

ki∏j=1

aji : i ∈ [n]

=∑

ν∈P(A)

∗b∈ν

κ (a ∈ b) ,

65

and

κ∗

ki∗j=1

aji : i ∈ [n]

=∑

ν∈P(A)

∏b∈ν

κ∗ (a ∈ b) ,

where the sums in both equalities run over all strongly-mixing partitions ofa multiset A = {aji : i ∈ [n], j ∈ [ki]}. Observe that in each equality thecumulants on each side are of the same type but the multiplications are not.In our formula we will mix types of cumulants on both sides but keep thesame multiplication.

To present our result we introduce a class of strongly-mixing forestsF (A).Definition 4.14. Let A1, . . . , An be multisets consisting of elements of A.Consider a reduced mixing forest F ∈ F (A) consisting of trees T1, . . . , Ts.Denote by av ∈ A the label of a leaf a ∈ A. We define a partition νF of aset A as follows:

νF ={{av : v ∈ Tk}k∈[s]

}.

We say that a mixing reduced forest F ∈ F (A) is strongly-mixing if thepartition νF is strongly-mixing partition. We denote the set of such forestsby F (A).Remark 4.15. Observe that the class of all strongly-mixing forests F (A) is asubclass of all mixing forests F (A), which itself is a subclass of all reducedforests F (A), i.e.:

F (A) ⊂ F (A) ⊂ F (A) .

This is analogue to the natural order between classes of strongly-mixingpartitions P (A), mixing partitions P (A) and partitions P (A):

P (A) ⊂ P (A) ⊂ P (A) .

We can reformulate Theorem 4.6 as follows.

Theorem 4.16 (Analogue of Leonov–Shiryaev’s formula). Consider an al-gebra A with two multiplicative structures · and ∗. Denote by κ and κ∗

cumulants related to the identity map on A as we described above. Then thefollowing formula holds:

κ∗

ki∗j=1

aji : i ∈ [n]

=∑

F∈F(A)

(−1)wF κF , (4.8)

where F (A) is a set consisting of strongly-mixing reduced forests.

66

Example 4.17. Figure 4.1 presents all reduced forests on the multiset A ={a1

1, a12, a

21}. Six of them are mixing, five of them are strongly mixing. Thus:κ∗(a1

1 ∗ a12, a

21)

= a11 ∗ a1

2 ∗ a21 − κ(a1

1, a21 ) ∗ κ(a1

2 )−κ(a1

2, a21 ) ∗ κ(a1

1 )− κ(a11, a

12, a

21 )

+κ(κ(a1

2, a22 ), κ(a1

1 )).

Proof. In (4.4) we present the moment cumulant formula for cumulants κrelated to the map (A, ·) id−→ (A, ∗). Similar expression for cumulants κ∗

related to the inverse map (A, ∗) id−1−→ (A, ·) is of the following form:

a1 · · · an =∑

ν∈P([n])∗b∈ν

κ∗ (ai : i ∈ b) .

We express the ·-product(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)via the moment

cumulant formula given by the equation above:(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑ν∈P([n])∗b∈ν

κ∗

ki∗j=1

aji : i ∈ [n]

. (4.9)

From Theorem 4.6 we can express the left-hand side of this equation inanother way:(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑F∈F(A)

(−1)wF κF .

Observe that we can split the summation of (−1)wF κF over all mixing re-duced forests F ∈ F (A) into ∗-product of summation over all strongly-mixing reduced forests:

∑F∈F(A)

(−1)wF κF =∑

ν∈P([n])∗b∈ν

∑F∈F(Ab)

(−1)wF κF

,where, for each partition ν, sets Ab := ∪i∈bAi are division of a set A.

Observe, that quantities: ∑F∈F(Ab)

(−1)wF κF

satisfy the system of equations given by the moment cumulant formula (4.9),which has a unique solution. This yields the statement of the theorem.

Remark 4.18. The above equation is still valid when we replace κ (which ishidden in κF terms) with κ∗ and replace ∗-products with ·-products simul-taneously.

67

4.3. Cumulant formula for Jack charactersIn this section we show that Jack characters forms an algebra with twomultiplications. In consequence, in Section 4.3.2 we present some statementsabout structure constants which may be seen via this characterisation of Jackcharacters.

4.3.1. Approximate factorization property

In many cases cumulants are quantities of a very small degree. The followingdefinition specifies this statement [Śn16, Definition 1.8].Definition 4.19. Let A and B be filtered unital algebras and let E : A −→ Bbe a unital linear map. Let κ be the corresponding cumulants. We say thatE has approximate factorization property if for all choices of a1, . . . , al ∈ Awe have that

degB κ (a1, . . . , al) ≤ degA a1 + · · ·+ degA al − 2 (l − 1) .

Observation 4.20. Let us go back to the case, when E is the identity mapon algebra A with two multiplications. Suppose that the identity map

(A, ·) id−→ (A, ∗) ,

satisfies the approximate factorization property.Let A1, . . . An be multisets consisting of elements of A. Let A be the sum

of those multisets. Then for any forest F ∈ F (A) consisting of f trees,there is the following restriction on the degree of cumulants:

deg κF ≤(∑a∈A

deg a)− 2|A|+ 2f,

where |A| is the number of elements in A.

Proof. We analyse the definition of κF (Definition 4.4). For any vertex v ∈ Fwe defined the quantities κv. Using the approximate factorization property,observe that:

deg κv ≤n∑i=1

deg κvi − 2 (n+ 1) .

Going from the root r to the leaves we obtain:

deg κr ≤nr∑i=1

deg κvi − 2 (nr + 1) .

68

where nr is the number of leaves in a tree rooted in r, and vi for i ∈ [nr] areleaves of this tree.

The cumulants κF were defined as follows:

κF := ∗iκVi ,

where Vi are roots of all trees in F , hence deg κF ≤∑

deg κVi . It is noweasy to see the statement of this observation.

4.3.2. Cumulants of Jack characters

Jack characters Chπ form a natural family (indexed by partitions π) offunctions on the set Y of Young diagrams. One can introduce two differentmultiplicative structures on the linear space spanned by Jack characters.

The ∗-product is given by concatenations of partitions:

Chπ ∗Chσ = Chπtσ .

For any partitions π and σ one can uniquely express the pointwise prod-uct of the corresponding Jack characters

Chπ ·Chσ =∑µ

gµπ,σ Chµ (δ)

in the linear basis of Jack characters.Śniady considers an algebra of Jack characters as a graded algebra, with

gradation given by the notion of α-polynomial functions [Śn15, Section 1.7].Jack characters are α-polynomial function of the following degrees

deg Chπ = |π|+ `(π).

Śniady gave explicit formulas for the top-degree homogeneous part of Jackcharacters. We sketch shortly how we use Theorem 4.6 in order to find thetop-degree coefficients of the structure constants below.

Consider two integer partitions π = (π1, . . . , πn) and σ = (σ1, . . . , σl)and the relevant multiset A = A1 ∪A2 given by:

A1 = {Chπ1 , . . . ,Chπn} ,A2 = {Chσ1 , . . . ,Chσl} .

Together with the ·-product and the ∗-product described above, the linearspace spanned by Jack characters becomes an algebra with two multiplica-tions. We can introduce cumulants as a way of measuring the discrepancybetween those two types of multiplications via (4.4). Recently the approxi-mation factorisation property of cumulants was proven [Śn16].

69

Lemma 4.21 (reformulation of Theorem 4.6). Let A1, . . . , An be multisetsconsisting of elements of A. Let A be the sum of those multisets. Then:

(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑ν∈P(A)

|ν|∗i=1

∑T∈T (νi)

(−1)wT κT .

where ∗ and · are two different multiplications on A and ν = {ν1, . . . , ν|ν|}is a partition of A.

Proof. Theorem 4.6 presents(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)as a sum

over reduced mixing forests of cumulants associated to those forests. Ob-serve that each reduced mixing forest F splits naturally into a collectionof trees T1, . . . , Tk. Each of Ti possesses the property of being reduced andmixing. Leaves of F are labelled by elements of A, thus we denoted by A themultiset consisting of those labels. Division of F into T1, . . . , Tk determinesa partition ν = {ν1, . . . , νk} of a set A, namely νi ⊂ A consists of all labelsof leaves of Ti. The cumulant κF is equal to:

k∗i=1

κTi

by the definition. Moreover (−1)wF =∏ki=1 (−1)wTi .

Theorem 4.22. With notation presented above, for any two partitions πand σ, the following decomposition is valid

Chπ ·Chσ =∑

ν∈P(A)

|ν|∗i=1

∑T∈T (νi)

(−1)wT κT , (4.10)

where ν = {ν1, . . . , ν|ν|} is a partition of A and T (νi) denotes the set of allreduced mixing trees on νi ⊆ A.

Moreover, there is the following restriction on the degree of products ofcumulants:

deg( |ν|∗i=1

∑T∈T (νi)

(−1)wT κT

)≤ |π|+ |σ|+ 2|ν|,

where |ν| is the number of parts in partition ν.

70

Presented statement is based on Lemma 4.21, the bound of a degreefollows immediately from Observation 4.20.

The division given in (4.10) is a tool for capturing the structure constantsgµπ,σ. It opens a way for induction over the number `(σ) + `(π). Moreprecisely, we express κT in in the linear basis of Jack characters inductively,according to the number of leaves.

4.4. Proof of Proposition 4.9In this section we shall prove Proposition 4.9. We use the same notation asin Section 4.1.4. We denote by A1, . . . , An multisets consisting of elementsof A. We denote by A = A1 ∪ · · · ∪An the multiset, which is the sum of allmultisets Ai. We use also the following notation for the elements of Ai:

Ai ={ai1, . . . , a

iki

},

hence the multiset A consists of the following elements:

A ={a1

1, . . . , a1k1 , . . . , a

n1 , . . . , a

nkn

}.

We denote additionally the set of all partitions of A by P (A). We denote byP (A) a set of all mixing partitions of A, i.e. all partitions ν = {ν1, . . . , νl}such that

∃i∈[l]

∀j∈[n]

νi 6⊆ Aj .

4.4.1. Outline of the proof

Firstly, we express the left-hand side of (4.6) as a sum of cumulants, wherethe sum runs over all mixing partitions ν ∈ P (A), see (4.11) below. Byapplying inductively the procedure (4.12) described below, we replace sum-mation over all mixing partitions ν ∈ P (A) by a sum over all nested upwardsequences of partitions, see the Definition 4.24. Then we construct a bijec-tion between such sequences and reduced forests F ∈ F (A) equipped withgap-free, weakly-mixing colourings c ∈ CF (see Definitions 4.3, 4.8). Lateron we will prove that the weighted sum over all gap-free colourings for afixed forest is either equal to 0 or to ±1.

4.4.2. Cumulants of mixing partitions

Observe that the following equality of the sets holds:

P (A) =(P (A1)× · · · × P (An)

)∪ P (A) ,

71

where the elements of the Cartesian product P (A1)× · · · × P (An) are un-derstood as a partition of a multiset A = A1 ∪ · · · ∪An.

We apply the moment-cumulant formula given in (4.4):

a11 ∗ · · · ∗ a1

k1 ∗ · · · ∗ an1 ∗ · · · ∗ ankn =

∑ν∈P(A)

κν .

We split all partitions P (A) into two categories: mixing partitions P (A)and products of partitions P (Ai). In this way:

∑ν∈P(A)

κν =∑

ν∈∐i∈[n] P(Ai)

κν +∑

ν∈P(A)

κν =n∏i=1

∑ν∈P(Ai)

κν +∑

ν∈P(A)

κν

=(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)+

∑ν∈P(A)

κν .

From the equations above we obtain the following formula:(a1

1 ∗ · · · ∗ a1k1

)· · ·(an1 ∗ · · · ∗ ankn

)=(a1

1 ∗ · · · ∗ a1k1

)∗ · · · ∗

(an1 ∗ · · · ∗ ankn

)−

∑ν∈P(A)

κν .

(4.11)

4.4.3. Cumulants of upward sequences of partitions

Each cumulant on the right-hand side of (4.11) is a ·-product of simplecumulants. We use the moment-cumulant formula in a form given below

a1 · · · ak = a1 ∗ . . . ∗ ak −∑

ν∈P([k])ν 6={{1},...,{k}}

κν(a1, . . . , ak ). (4.12)

to replace ·-products by ∗-products and ·-products consisting of a strictlysmaller number of components.

For each cumulant on the right-hand side in (4.11) we apply the proce-dure (4.12). As an output we get one term which is a ∗-product of cumulantsand several terms of the form of a ·-product of cumulants. Observe that ineach term of the second type the number of factors is strictly smaller thanbefore applying the procedure. We apply to them this procedure iterativelyas long as we have ·-terms in our extension. In the end we get a sum of theterms given by ∗-product and cumulants.

72

Example 4.23. Let us express(a1

1 ∗ a22)· a2

1 using the procedure describedabove:(

a11 ∗ a1

2)· a2

1(4.11)= a1

1 ∗ a12 ∗ a2

1 − κ(a11, a

12, a

21 )

−κ(a11, a

21 ) · κ(a1

2 )− κ(a12, a

21 ) · κ(a1

1 )

(4.12)= a11 ∗ a1

2 ∗ a21 − κ(a1

1, a12, a

21 )

−κ(κ(a1

1, a21 ), κ(a1

2 ))

+ κ(κ(a1

2, a21 ), κ(a1

1 ))

−κ(a11, a

21 ) ∗ κ(a1

2 ) + κ(a12, a

21 ) ∗ κ(a1

1 ).

To formalize our idea we define nested upward sequences and theirs cu-mulants.Definition 4.24. A sequence of partitions ω =

(ν1 ↗ · · · ↗ νr

)is said to be

upward ifνi+1 is a partition of the set νi,

for any 1 ≤ i ≤ r − 1 and ν1 is a partition of a multiset A. Moreover,if for each i the partition νi+1 is non-trivial, i.e. νi+1 6=

{νi}, it is said

to be nested. We define the length of an upward sequence of partitionsω =

(ν1 ↗ · · · ↗ νr

)as the length of a sequence, and we denote |ω| = r.

Let us provide a simple example.Example 4.25. Consider a 5-element multiset A = {a1, . . . , a5} and the fol-lowing nested upward sequences of partitions ω1 = (ν1 ↗ ν2) and ω2 =(ν1 ↗ ν2 ↗ ν3), where:

ν1 ={{a1, a4

},{a2},{a3},{a5}},

ν2 ={{{

a1, a4},{a2},{a3}},{{a5}}}

,

ν3 ={{{{

a1, a4},{a2},{a3}},{{a5}}}}

.

We introduce the following technical notation (similar to the definitionof a cumulant κν for a partition ν).

κν{a1, . . . , an} :={κ(ai1 , . . . , ai|νj | )

}lj=1.

73

Definition 4.26. Let ν be a partition of a multiset A = {a1, . . . , an}. Con-sider an upward sequence of partitions ω =

(ν1 ↗ · · · ↗ νr

)such that

ν1 = ν.

We define the cumulant associated to the sequence ω as follows

κω := ∗b ∈ κνr−1

(...κν1

(a1,··· ,an

)) b.Example 4.27. The cumulants κω1 and κω2 associated to the nested upwardsequences of partitions ω1 and ω2 respectively from Example 4.25 are of thefollowing forms:

κω1 = κ(κ(a1, a4 ), κ(a2 ), κ(a3 )

)∗ κ(κ(a5 )

)= κ

(κ(a1, a4 ), a2, a3

)∗ a5,

κω2 = κ

(κ(κ(a1, a4 ), κ(a2 ), κ(a3 )

), κ(κ(a5 )

))

= κ

(κ(κ(a1, a4 ), a2, a3

), a5

),

where we used the property κ(x) = x.Definition 4.28. Consider a multiset A = A1 ∪ · · · ∪ An. Denote by N (A)the set of all nested upward sequences of partitions ω =

(ν1 ↗ · · · ↗ νr

)such that ν1 ∈ P (A) is a mixing partition.

Proposition 4.29. Consider a multiset A = A1 ∪ · · · ∪An. Then∑ν∈P(A)

κν = −∑

ω∈N (A)(−1)|ω|κω. (4.13)

Proof. Apply procedure (4.12) iteratively to the left-hand side of Proposi-tion 4.29. Observe that applying this iterative procedure is nothing else butsumming over all nested upward sequences of partitions. The sign of theterm is determined by the number of iterations. Partition ν1 describes thefirst application of the procedure (this is why ν1 ∈ P (A)), partition ν2 thesecond, and so on.

Observe that different nested upward sequences of partitions ω maylead to the same cumulant κω. The following example illustrates this phe-nomenon.

74

Example 4.30. Let A1 = {a11, a

12} and A2 = {a2

1, a22}. Consider ω1, ω2, ω3 ∈

N (A) given by ω1 =(ν1

1 ↗ ν21)and ω2 =

(ν1

2 ↗ ν22 ↗ ν3

2)and ω3 =(

ν13 ↗ ν2

3 ↗ ν33), where:

ν11 =

{{a1

1, a22},{a1

2, a21}}, ν2

1 ={{{

a11, a

22},{a1

2, a21}}}

,

ν12 =

{{a1

1, a22},{a1

2},{a2

1}}, ν2

2 ={{{

a11, a

22}},{{a1

2},{a2

1}}}

,

ν32 =

{{{{a1

1, a22}},{{a1

2},{a2

1}}}}

,

ν13 =

{{a1

1},{a2

2},{a1

2, a21}}, ν2

3 ={{{

a11},{a2

2}},{{a1

2, a21}}}

,

ν33 =

{{{{a1

1},{a2

2}},{{a1

2, a21}}}}

.

Observe that all sequences ω1, ω2, ω3 lead to the same term up to the sign.Moreover, they are the only ones which lead to this cumulant. Observe that

(−1)|ω1|κω1 + (−1)|ω2|κω2 + (−1)|ω3|κω3 = κ(κ(a1

1, a22 ), κ(a1

2, a21 )).

With the weights given by (−1)|ωi|, cumulants corresponding to sequencesω1, ω2, ω3 sum up to just one term. We will see that this is true in general.

4.4.4. Reduced forests and their colourings

To each upward nested sequence of partitions ω =(ν1 ↗ · · · ↗ νr

)we shall

assign a certain rooted forest with a colouring. We construct a bijectionbetween the sequences from N (A) and relevant rooted forests equippedwith the colourings.Definition 4.31. Let ω =

(ν1 ↗ · · · ↗ νr

)be a nested sequence of par-

titions. Denote the elements of partition νi by νi = {νi1, . . . , νiki}. Letν1 = {ν1

1 , . . . , ν1k−1} be a partition of A = A1 ∪ · · · ∪ An. We associate to ω

a rooted forest with coloured vertices by the following procedure:

• The elements of A are leaves of the forest. We colour each of them by0.

75

a11 a1

2 a21 a2

2

1 1

2

a11 a1

2 a21 a2

2

1

2

3

a11 a1

2 a21 a2

2

2

1

3

Figure 4.3 – The forests associated with ω1, ω2, ω3 from Example 4.30. Allleaves are coloured by 0.

• For each element νij , where 1 ≤ i ≤ r and 1 ≤ j ≤ ki, we create avertex and colour it by i.

• We join νi−1j and νij if νi−1

j ⊆ νij . Similarly we join a ∈ A and ν1j if

a ∈ ν1j .

• We delete each vertex v which has only one descendant. We join thedescendant and the parent of v.

We denote by Φ1(ω) the forest and by Φ2(ω) the colouring associated to ω.Example 4.32. The coloured forests associated with ω1, ω2, ω3 from Ex. 4.30are presented on Figure 4.3. Since ω1, ω2, ω3 start from one element parti-tion, all three forests are trees.

The forest described in Definition 4.31 consists of kr rooted trees, wherekr is a number of elements in νr, namely νr = {νr1 , . . . , νrkr}. The conditionof nestedness of ω translates to the fact that each colour is used. Except forleaves, each vertex has at least two descendants. It leads to the definition ofreduced forest and gap-free, weakly-mixing colouring. We mentioned theirdefinitions in the introduction (see Definitions 4.3 and 4.8).

Lemma 4.33. There exists a bijection Φ between the set N (A) of nestedupward sequences starting with a mixing partition and the set of pairs (F, c)consisting of a reduced forest F ∈ F (A) of length r ≥ 1 with a gap-free,weakly-mixing colouring c ∈ CF :

Φ : ω 7−→ (F, c) := (Φ1(ω),Φ2(ω))

For any nested upward sequence starting with a mixing partition ω ∈N (A), the following equality of cumulants holds

κω = κΦ1(ω),

76

where κΦ1(ω) is a cumulant of a reduced forest Φ1(ω), see Definition 4.4.Moreover |ω| = |Φ2(ω)| i.e. the length of the nested upward sequence is

equal to the length of the corresponding colouring.

Proof. Definition 4.31 shows already how to associate a reduced forest F :=Φ1(ω) with the gap-free colouring c := Φ2(ω) to a nested upward sequenceω. The construction is done in such a way that |c| = |ω|. For the reversedirection, the algorithm is easily reproducible. The condition that a nestedupward sequence ω =

(ν1 ↗ · · · ↗ νr

)∈ N (A) starts with a mixing par-

tition ν1 ∈ P (A) translates to the condition of c being a weakly-mixingcolouring (Definition 4.8).

In Definition 4.4 we introduced cumulant κF for a forest F ∈ F (A).There is an exact correspondence between this expression and the one, whichis given in Definition 4.26.

We are ready to prove Proposition 4.9 which is the purpose of this sec-tion. Let us recall its statement:

Proposition 4.9. Let A1, . . . , An be multisets consisting of elements of A.Let A be a sum of those multisets. Then(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=

∑F∈F(A)

κF∑c∈CF

(−1)|c|. (4.6)

Proof. Combining the formula (4.11) and the Proposition 4.29 lead to thefollowing expression:(

a11 ∗ · · · ∗ a1

k1

)· · ·(an1 ∗ · · · ∗ ankn

)=(

a11 ∗ · · · ∗ a1

k1

)∗ · · · ∗

(an1 ∗ · · · ∗ ankn

)+

∑ω∈N (A)

(−1)|ω|κω.

We identify the product term on the right-hand side of the equationabove with the only reduced forest of length r = 0. Indeed, there is justone reduced forest of length r = 0 and the only one gap-free, weakly-mixingvertex colouring c of it, namely the forest F consisting of separated verticesa ∈ A, each coloured by 0. The term

(a1

1 ∗ · · · ∗ a1k1

)∗ · · · ∗

(an1 ∗ · · · ∗ ankn

)is equal to the corresponding cumulant κF .

We replace the sum term on the right-hand side of the equation above,according to the bijection between sequences ω ∈ N (A) and reduced forestsof length r ≥ 1 with gap-free, weakly-mixing colourings given in Lemma4.33.

77

4.5. Proof of Proposition 4.10In this section we shall prove Proposition 4.10. For a given reduced forest F ,we investigate the following sum∑

c∈CF

(−1)|c|

over all gap-free, weakly-mixing colourings of F , which occur in Proposi-tion 4.9.

4.5.1. Parameter wF of a reduced forest F ∈ F (A)

We introduce an invariant wF which determines the coefficient of κF . Thisdefinition was already mentioned in Section 4.1.4, we recall it below andnext extend it slightly:Definition 4.5. Consider a multiset A = A1 ∪ · · · ∪An and a reduced forestF ∈ F (A). We say that F is mixing for a division A1, . . . , An (or shortlymixing) if for each vertex v whose descendants are all leaves, those descen-dants are elements of at least two distinct multisets Ai and Aj . Denote byF(A) the set of all reduced mixing forests.

For a reduced mixing forest F we define the quantity wF to be thenumber of vertices in F minus the number of leaves (see Figure 4.1).

If F is not mixing, we define wF :=∞. We may also introduce numberwF inductively, according to the height of a forest.Definition 4.34. Let F be a reduced forest. The height of a forest F is themaximum distance between one of the roots and one of its leaves. We denotethis quantity by h(F ).Definition 4.35 (Definition equivalent to Definition 4.5). Let F be a reducedforest and F1, . . . , Fr its sub-forests obtained by deleting the roots of F . Wedefine the number wF ∈ N ∪ {∞} inductively on h(F ) as follows:

wF =

∑ri=1 kFi + 1 if h(F ) ≥ 2,∞ if h(F ) = 1 and all descendants of some root

belong to just one multiset Ai for some i ∈ [n],1 if h(F ) = 1 and for each root there are at least

two descendants belonging to some two distinctmultisets Ai and Aj ,

0 if h(F ) = 0.

78

a11 a1

3 a23

a21 a2

2 a31v1

a32 a1

2 a34

wT =∞ wT = 6

a11 a1

4 a12

a21 a2

2 a13v3

v2a3

2 a15 a1

6

Figure 4.4 – The tree on the left-hand side is mixing. Indeed, descendantsof the vertex v1 belong to at least two multisets: A1 and A3. Observe thatthe tree on the right-hand side is not mixing. Indeed, there are two verticesof height equal to one: v2 and v3. All descendants of v2 belong to A2 andall descendants of v3 belong to A1.

Example 4.36. Let A = A1 ∪ A2 ∪ A3. On Figure 4.4, we give an exampleof two forests (in particular trees) and we count the two corresponding wFnumbers. Observe that number wF depends on the labels of the leaves inthe forest F .

4.5.2. Proof of Proposition 4.10

Let us recall the statement of proposition.

Proposition 4.10. Let A1, . . . , An be multisets consisting of elements of A.Let A be a sum of those multisets. Then, for any reduced forest F ∈ F (A),the following holds:

∑c∈CF

(−1)|c| ={

(−1)wF if F ∈ F (A) ,0 otherwise.

Proof of Proposition 4.10 is divided into two cases: either a forest Fis not mixing, i.e. wF = ∞ (Lemma 4.37), or a forest F is mixing, i.e.wF 6=∞ (Lemma 4.38). The next two subsections establish these two cases.

4.5.3. Proof of the not mixing case

Lemma 4.37. Let A1, . . . , An be multisets consisting of elements of A. LetA be a sum of those multisets. For any reduced forest F witch is not mixing,

79

we have ∑c∈CF

(−1)|c| = 0.

Proof. Since F is not mixing, there exists a vertex v such that all of itsdescendants are leaves, and all of them belong to just one multiset Ai forsome i ∈ [n]. Consider the following partition of set CF :{

Cki}k=1,2

i∈Z

where each C1i consists of all c ∈ CF with |c| = i and where the vertex v is

coloured by its own colour; C2i consists of all c ∈ CF with |c| = i and where

there is another vertex coloured by the same colour as the vertex v. Weexpress the sum over all c ∈ CF as follows:

∑c∈CF

(−1)|c| =∑i∈Z

∑c∈C1

i

(−1)i +∑c∈C2

i

(−1)i =

∑i∈Z

(−1)i∑c∈C1

i

1−∑

c∈C2i−1

1

.We will show the equipotency of the sets C1

i and C2i−1 from which it follows

that the sum above is equal to 0 and the statement of the lemma is true.Let us construct a bijection between C1

i and C2i−1. Take any c ∈ C1

i .Suppose that the vertex v is coloured by k. Observe that k ≥ 2. Indeed, if vwas coloured by 1, it would be the only vertex of this colour. Then, the only1-coloured vertex would have descendants belonging to just one multiset Ai,which is in contradiction with the fact that c ∈ CF (i.e. c is a weakly-mixingcolouring). From c ∈ C1

i we construct c′ ∈ C1i−1 as follows:

1. keep the colours of vertices coloured by 1, . . . , k − 1 unchanged,

2. change the colours of vertices coloured by k, . . . , i to k − 1, . . . , i − 1respectively.

This procedure is reversible. Indeed, take c′ ∈ C1i−1 and suppose that vertex

v is coloured by k for some k ≥ 1. Then c ∈ C2i can be recovered by the

following procedure:

1. do not change colours of the vertices coloured by 1, . . . , k − 1,

80

2. do not change the colour of v,

3. change the colours of the vertices coloured by k, . . . , i−1 to k+1, . . . , irespectively (excluding vertex v).

4.5.4. How to prove the mixing case?

We will prove the following lemma.

Lemma 4.38. Let A1, . . . , An be multisets consisting of elements of A. LetA be a sum of those multisets. For any reduced mixing forest F , we have:∑

c∈CF

(−1)|c| = (−1)wF .

To prove the lemma above we show a bijection between gap-free colour-ings of reduced trees T ∈ T (A) and gap-free colourings of reduced forestsF ∈ F (A), which are not trees (see Remark 4.39). Using this bijectionwe can restrict proof of Lemma 4.38 just to trees. For reduced trees andtheir gap-free colourings we define a projection of this colourings (see Defi-nition 4.40). We make use of the notion of projection in Lemma 4.41. Proofof Lemma 4.38 is done by induction on number of vertices in tree T andpresented in Section 4.5.7.

4.5.5. Restriction to the trees

Remark 4.39. There is a natural bijection f between all reduced trees T ∈T (A) and all reduced forests F ∈ F (A), which are not trees. This bijectionis obtained by deleting the root of T (see Figure 4.1). Moreover, for a givenreduced tree T ∈ T (A), there is an obvious bijection fT between all gap-free colourings of T and all gap-free colourings of the corresponding reducedforest f(T ), obtained by keeping the colours of the non-deleted vertices, sothat

|fT (c)| = |c| − 1.

Additionally fT preserves the property of being a weakly-mixing colouring.The above statement allows us to prove Lemma 4.38 just for the case

of trees T ∈ T (A) and conclude the statement for all forests F ∈ F (A).Indeed, suppose that the statement of Lemma 4.38 holds for trees. Con-sider a mixing forest F ∈ F (A) which is not a tree. Then, the treeT := f−1(F ) ∈ T (A) is also mixing, hence we can use the statement of

81

Lemma 4.38. Observe that wF = wT − 1. Using bijections f and fT we getthe following equality

∑c∈CF

(−1)|c| =∑

c∈Cf−1(F )

(−1)|f−1T (c)|+1 =

−∑c∈CT

(−1)|c| = − (−1)wT = (−1)wF ,

which is the statement of Lemma 4.38 for the mixing forest F ∈ F (A).

4.5.6. Projection of a gap-free colouring

For any reduced tree T ∈ T (A) we consider sub-trees T1, . . . , Tk formedby deleting the root of T . Number k is equal to the degree of the root.Every sub-tree Ti is also a reduced tree. Every gap-free colouring c inducesalso a sub-colourings c1, . . . , ck on T1, . . . , Tk. Observe that sub-colouringsobtained this way are not necessarily gap-free. However, there is a canonicalway to make them gap-free.Definition 4.40. Let T be a reduced tree with a gap-free colouring c. Letc1, . . . , ck be the induced colourings on sub-trees T1, . . . , Tk formed by delet-ing the root of T . For some i ∈ [k], let ji0 < · · · < jil be the sequence ofcolours used in the colouring ci. By replacing each jin by n in the colouring ciwe obtain a gap-free colouring, which we denote by ci. We say that ci as ani-th projection of the colouring c and denote it as pi(c) := ci, see Figure 4.5.

Lemma 4.41. Let T be a reduced mixing tree of height h(T ) ≥ 2. Denoteby T1, . . . , Tr all sub-trees obtained by deleting the root of T . Let c1, . . . , crbe gap-free colourings of T1, . . . , Tr respectively. Then the following equalityholds: ∑

c∈CTpi(c)=ci

(−1)|c| = −r∏i=1

(−1)|ci|.

Proof of Lemma 4.41. Observe that if T is the mixing tree, then any gap-free colouring c belongs to CT . Indeed, take any vertex v coloured by 1.Clearly, its descendants are leaves labelled by elements of at least two distinctmultisets Ai and Aj (by assumption that T is mixing). The existence of sucha vertex implies that c ∈ CT .

The proof is divided into three steps: first, we construct a bijection be-tween the gap-free colouring c projecting onto c1, . . . , cr and some integer

82

1

2 3

44

5

1

2 3

4

T2

4

T1

1

1 2

3

T2

2

T1

T1

T21 2 3

1

2

(a) Colouring c of a tree T usesthe colours {0, 1, 2, 3, 4, 5}.

(b) Colouring c1 of the tree T1 usesthe colours {0, 2, 4}. Colouringc2 of the tree T2 uses the colours{0, 1, 3, 4}.

(c) Projections p1(c) andp2(c) use colours {0, 1, 2} and{0, 1, 2, 3} respectively.

(d) Path ρ constructed from c inproof of Lemma 4.41.

Figure 4.5 – (a) A reduced tree T with a gap-free colouring c. (b) Bydeleting the root we obtain two reduced subtrees: T1 and T2 with inheritedcolourings: c1 and c2. Observe that they are not gap-free. (c) However, theprocedure given in Definition 4.40 describes the canonical way of producinga gap-free colourings p1(c) and p2(c). (d) Moreover, for the colouring c wepresent an associated path ρ which will be introduced in a proof of Lemma4.41.

83

paths in Nr; second, we introduce a generating function of these paths andcharacterise it by recursion on the endpoints and some boundary condition;finally, we find a function satisfying those conditions.

Step 1. Let us recall that any gap-free colouring c of T induces colouringsci on Ti, from which we deduce a gap-free colouring ci of Ti (Definition4.40). We shall construct a bijection between all gap-free colourings c of Tprojecting on c1, . . . , cr and all integer paths ρ such that:

• ρ connects (0, . . . , 0) and (|c1|, . . . , |cr|) ∈ Nr,

• each step of ρ is of the following form: (kn1 , . . . , knr ) ∈ {0, 1}r\(0, . . . , 0) .

Denote the class of such paths by P|c1|,...,|cr|. Moreover the construction isdone in such a way that |c| = |ρ|+ 1, where by |ρ| we denote the number ofsteps in ρ.

For a gap-free colouring c we construct a path ρ starting from (0, . . . , 0) ∈Nr by the following procedure: the n-th step of ρ is of the form (kn1 , . . . , knr )where:

kni ={

1 if colour n appears in ci,0 if it does not.

An example of such path is presented on Figure 4.5.The procedure described above is reversible. Indeed, take a path ρ be-

tween (0, . . . , 0) and (|c1|, . . . , |cr|) ∈ Nr. Suppose that the n-th step is ofthe form:

(kn1 , . . . , knr ) ∈ {0, 1}r \ (0, . . . , 0) .We can assign to the path ρ a colouring c by the following procedure. Let(x1, . . . , xr) be an endpoint of ρ after the n-th step. We colour each vertexv ∈ Fi by n if v was coloured by xi in colouring ci and kni 6= 0. We colourthe root by |ρ|+ 1.

Step 2. The bijection from Step 1 was constructed in such a way that|c| = |ρ|+ 1. Observe that∑

c∈CTpi(c)=ci

(−1)|c| =∑

ρ∈P|c1|,...,|cr |

(−1)|ρ|+1 .

Let us define a function F : Zr −→ Z:

F : (x1, . . . , xr) 7−→∑

ρ∈Px1,...,xr

(−1)|ρ|+1 .

84

Observe that:

I. for all (x1, . . . , xr) 6∈ Nr, F (x1, . . . , xr) = 0,

II. F (0, . . . , 0) = −1,

III. for all (x1, . . . , xr) ∈ Nr\(0, . . . , 0), the function F satisfies the followingrecursive formula:

F (x1, . . . , xr) = −∑X⊆[r]X 6=∅

F (xX1 , . . . , xXr ),

where xXi ={

xi if i 6∈ Xxi−1 if i ∈ X

Let us shortly comment on this observation. There are no paths con-necting (0, . . . , 0) with points (x1, . . . , xr) 6∈ Nr using the set of steps whichis non-negative (Observation I). There is just one path connecting point(0, . . . , 0) to itself: the empty path. Its length is equal to 0 (ObservationII). Consider all possibilities for the last step in path ρ. It is equivalentto choosing indices X ⊂ [r], X 6= ∅ and summing over all paths endingin (xX1 , . . . , xXr ) multiplied by −1, because we count the sign of the path(Observation III).

Those three statements about function F define it uniquely. The recur-sive formula gives us the way to compute F (x1, . . . , xr) inductively accordingto

r∑i=1

xi. The first and the second observation give us the starting point for

our induction, namely the values F (x1, . . . , xr) forr∑i=1

xi = 0.

Step 3. We show now that the function G : Zr −→ Z:

G : (x1, . . . , xr) 7−→{−∏ri=1 (−1)xi if for all i : xi ≥ 0,

0 otherwise.

satisfies all three properties I, II, III mentioned in Step 2. Hence, those twofunctions F and G are equal. By connecting the results of each step, we getthe statement of the lemma:∑

c∈CTpi(c)=ci

(−1)|c| =∑ρ∈P(|c1|,...,|cr |)

(−1)|ρ|+1 = F (|c1|, . . . , |cr|) =

= G(|c1|, . . . , |cr|) = −∏ri=1 (−1)|ci| .

We shall show that function G satisfies three properties I, II, III. Clearly,it satisfies I, II. In order to show that the recursive formula also holds, take

85

any (x1, . . . , xr): ∀ixi ≥ 0 and

r∑i=1

xi > 0.

Define the set Y consisting of boundary indices i of the point (x1, . . . , xr).More precisely, define Y ⊂ [r] as follows:

{i∈Y if xi > 0i 6∈Y if xi = 0 .

In order to show that G satisfies III, we have to show the vanishing ofthe following sum:

G(x1, . . . , xr) +∑X⊂[r]X 6=∅

G(xX1 , . . . , xXr ) =∑X⊂[r]

G(xX1 , . . . , xXr ) =

=∑X⊂Y

G(xX1 , . . . , xXr ) +∑X 6⊂Y

G(xX1 , . . . , xXr ).(4.14)

Observe that summants of the sum over X 6⊂ Y are equal to 0. FromX 6⊂ Y it follow that there exists i ∈ [r] such that i ∈ X and i 6∈ Y . Itmeans that xXi = −1 and by definition G(xX1 , . . . , xXr ) = 0.

Observe that the summants in the sum over X ⊂ Y are of the form−∏ri=1 (−1)x

Xi . Indeed, from X ⊂ Y it follows that xXi ≥ 0 for all i ∈ [r].

We use this observation to show the vanishing of the sum in (4.14):

(4.14) =∑X⊂Y

−r∏i=1

(−1)xXi = −

∑X⊂Y

(−1)

r∑i=1

xi−|X|

= (−1)

r∑i=1

xi·|Y |∑i=0

(|Y |i

)· (−1)i = 0.

4.5.7. Proof of Lemma 4.38

Remark 4.42. Let T be a reduced mixing tree such that h(T ) ≤ 2. LetT1, . . . , Tr be sub-trees obtained from F by deleting the roots. For anycolouring c ∈ CF the projection ci := pi(c) is in CTi for each i ∈ [r].

Indeed, by definition, ci = pi(c) are gap-free colourings of Fi. Take anyvertex v coloured by 1 in ci. Its descendants are leaves. Hence wF 6= ∞,so also kFi 6= ∞. That means that descendants of v belong to at least twodistinct multisets Ai. The existence of such vertex implies that c ∈ CTi .

Proof of Proposition 4.38. We will use induction on the height of tree T ∈T (A).

86

We cannot begin with a tree of height 0, namely a one-vertex tree T = •,because there is no such s tree if |A| ≥ 2.

Induction base. We begin from a tree T of height one, namely consistingjust of the root and leaves. We have exactly one gap-free colouring of length1, namely leaves are coloured by 0 and the root by 1. The claim followsimmediately.

Induction step. Let n ≥ 2. Suppose now that the statement of Lemma4.38 is true for any tree T of the height h(T ) ≤ n−1. We will show that it isalso true for any tree of height equal to n ≥ 2. Take such a tree T . Denoteby T1, . . . , Tr its sub-trees obtained from T by deleting the root. Clearlyfor every Ti, the h(Ti) ≤ n− 1 and we can use the induction hypothesis forthem. We have:

∑c∈CT

(−1)|c| Remark 4.42=∑

c1∈CT1···cr∈CTr

∑c∈CT

pi(c)=ci

(−1)|c|

Lemma 4.41= −∑

c1∈CT1···cr∈CTr

r∏i=1

(−1)|ci| = −r∏i=1

∑ci∈CTi

(−1)|ci|

Induction= −r∏i=1

(−1)kTi

Definition 4.35= (−1)kT ,

which proves the statement for any tree of height equal to n.

87

Appendix A

Top-degree parts in theMatchings-Jack Conjectureand the b-Conjecture

We shall prove that our result about the top-degree part in the Matchings-Jack Conjecture presented in Theorem 2.5 and the result of Dołęga [Doł17c,Theorem 1.5] about the top-degree part in b-Conjecture are equivalent.

First note that the polynomials cλπ,σ and hλπ,σ are related as follows

∑n≥1

tn∑

λ,π,σ`nhλπ,σpπ(x)pσ(y)pλ(z) =

αt∂

∂tlog

∑n≥1

tn∑

λ,π,σ`n

cλπ,σα`(λ)zλ

pπ(x)pσ(y)pλ(z)

(A.1)

and

∑n≥1

tn∑

λ,π,σ`n

cλπ,σα`(λ)zλ

pπ(x)pσ(y)pλ(z) =

exp

∑n≥1

1αn

tn∑

λ,π,σ`nhλπ,σpπ(x)pσ(y)pλ(z)

, (A.2)

see (2.1) and (2.2).

88

In Theorem 2.5 we showed that the leading coefficient of cλπ,σ can beexpressed in the following way:[

βd(π,σ;λ)]cλπ,σ =

∣∣∣M ∈Mλ;λπ,σ : M is unhandled

∣∣∣whereMλ;λ

π,σ is the set of λ-lists of unicellular maps with the white and blackvertices distribution given by π and σ respectively.

On the other hand, Dołęga [Doł17c, Theorem 1.5] showed that the lead-ing coefficient of h(n)

π,σ can be expressed in the following way:[βd(π,σ;(n))

]h(n)π,σ =

∣∣∣M ∈M (n)π,σ : M is unhandled

∣∣∣where M (n)

π,σ is the set of unicellular maps with the white and black verticesdistribution given by π and σ respectively.Remark A.1. Observe that multiplication of power-sum symmetric functionsexpresses as follows

pλ1(z) · pλ1(z) = pλ1∪λ1(z)

in the terms of concatenations of relevant partitions.

We investigate the[p(n)(z)

]coefficient in both sides of (A.1). We have

tn∑π,σ`n

h(n)π,σpπ(x)pσ(y) = αt

∂

∂t

tn ∑π,σ`n

c(n)π,σ

αz(n)pπ(x)pσ(y)

=

αntn∑π,σ`n

c(n)π,σ

αnpπ(x)pσ(y)

hence c(n)π,σ and h(n)

π,σ are equal.Since c(n)

π,σ = h(n)π,σ, it might seem that our result extends the result of

Dołęga. However, a more subtle analysis of relationships between the coef-ficients of cλπ,σ and hλπ,σ shows that both results are equivalent.

The power series expansion of the exponent function in (A.2) gives us

∑n≥1

tn∑

λ,π,σ`n

cλπ,σα`(λ)zλ

pπ(x)pσ(y)pλ(z) =

∑k≥0

1k!

∑s≥1

1sts

∑λ,π,σ`s

hλπ,σα

pπ(x)pσ(y)pλ(z)

k . (A.3)

89

We denote by Pλ,π,σk the set of triplets of lists of partitions( (λ1, . . . , λk

),(π1, . . . , πk

),(σ1, . . . , σk

) )such that

k⋃i=1

λi = λ,k⋃i=1

µi = µ,k⋃i=1

σi = σ

and for each i we have |λi| = |πi| = |σi|.Let us investigate the [pπ(x)pσ(y)pλ(z)] coefficient in both sides of (A.3).

We have

tncλπ,σ

α`(λ)zλ= tn

∑k:1≤k≤`(λ)

1k!

∑((λ1,...,λk),(π1,...,πk),(σ1,...,σk)

)∈Pλ,π,σ

k

k∏i=1

1|λi|

hλi

πi,σi

α. (A.4)

Dołęga and Féray [DF17, Theorem 1.2] gave the following bound on thedegree

deg hλiπi,σi ≤ |λi|+ 2− `(λ1)− `(πi)− `(σi).

Hence, each summand of the first sum on the right-hand side of (A.4) hasdegree equal to at most

n+ k − `(λ)− `(π)− `(σ),

and the maximal bound may be achieved only for summands correspondingto k = `(λ). For such a summand, its bound on the degree is the same asthe bound on the degree for the left-hand side of (A.4) given by (2.3). Wehave

1zλ

[αd(π,σ;λ)

]cλπ,σ =

1`(λ)!

∑((λ1,...,λ`(λ)),(π1,...,π`(λ)),(σ1,...,σ`(λ))

)∈Pλ,π,σ

`(λ)

`(λ)∏i=1

1|λi|

[α|λi|+1−`(π)−`(σ)

]hλiπi,σi

.

For a partition λ = (λ1, . . . , λk), denote by Cλ the set of all compositionsof a type λ, i.e. the set of all lists (λσ(1), . . . , λσ(k)), for some σ ∈ S (n).Observe that

|Cλ| =`(λ)!∑imi(λ)! .

90

Observe that for k = `(λ) the first list in any triplet from Pλ,π,σk is a com-position of a type of the Young diagram λ. We have

1zλ

[αd(π,σ;λ)

]cλπ,σ =

1`(λ)! |Cλ|

∑(((λ1),...,(λ`(λ))

),(

π1,...,π`(λ)),(

σ1,...,σ`(λ)))∈Pλ,π,σ

`(λ)

`(λ)∏i=1

1λi

[αλi+1−`(π)−`(σ)

]h

(λi)πi,σi

and hence

[αd(π,σ;λ)

]cλπ,σ =

∑(((λ1),...,(λ`(λ))

),(

π1,...,π`(λ)),(

σ1,...,σ`(λ)))∈Pλ,π,σ

`(λ)

`(λ)∏i=1

[αλi+1−`(π)−`(σ)

]h

(λi)πi,σi

Dołęga’s result [Doł17c, Theorem 1.5] shows us that[αλi+1−`(π)−`(σ)

]h

(λi)πi,σi

=∣∣∣M ∈M (λi)

πi,σi: M is unhandled

∣∣∣.Directly from the definition of Pλ,π,σ`(λ) we obtain that

[αd(π,σ;λ)

]cλπ,σ =

∣∣∣M ∈Mλ;λπ,σ : M is unhandled

∣∣∣,which allows us to conclude the equivalence of both results.

91

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