On the matchings-Jack and hypermap-Jack conjectures for labelled matchings and star hypermaps

# On the matchings-Jack and hypermap-Jack conjectures for labelled matchings and star hypermaps

## Abstract

Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack conjecture and the hypermap-Jack conjecture (also known as the -conjecture) are two major open questions relating Jack symmetric functions, the representation theory of the symmetric groups and combinatorial maps. They show that the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions and in the logarithm of the same sum interpolate respectively between the structure constants of the class algebra and the double coset algebra of the symmetric group and between the numbers of orientable and locally orientable hypermaps. They further provide some evidence that these two families of coefficients indexed by three partitions of a given integer and the Jack parameter are polynomials in with non negative integer coefficients of combinatorial significance. This paper is devoted to the case when one of the three partitions is equal to . We exhibit some polynomial properties of both families of coefficients and prove a variation of the hypermap-Jack conjecture and the matchings-Jack conjecture involving labelled hypermaps and matchings in some important cases.

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## 1 Introduction

### 1.1 Cauchy sums for Jack symmetric functions

For any integer denote an integer partition of with parts sorted in decreasing order. The set of all integer partitions (including the empty one) is denoted . If is the number of parts of that are equal to , then we may write as and define , . When there is no ambiguity, the one part partition of integer , is simply denoted . Given a parameter , denote and the power sum and Jack symmetric function indexed by on . Jack symmetric functions are orthogonal for the scalar product defined by . Denote the value of the scalar product This paper is devoted to the study of the following series for Jack symmetric functions introduced by Goulden and Jackson in [GouJac96].

 Φ(x,y,z,t,α) =∑γ∈PJαγ(x)Jαγ(y)Jαγ(z)t|γ|⟨Jαγ,Jαγ⟩α, Ψ(x,y,z,t,α) =αt∂∂tlogΦ(x,y,z,t,α).

More specifically, we focus on the coefficients and in their power sum expansions defined by:

 Φ(x,y,z,t,α) =∑n⩾0tn∑λ,μ,ν⊢nα−ℓ(λ)z−1λaλμ,ν(α)pλ(x)pμ(y)pν(z), Ψ(x,y,z,t,α) =∑n⩾0tn∑λ,μ,ν⊢nhλμ,ν(α)pλ(x)pμ(y)pν(z).

Goulden and Jackson conjecture that both the and may have a strong combinatorial interpretation. In particular thanks to exhaustive computations of the coefficients they show that the and are polynomials in with non negative integer coefficients and of degree at most for all . They conjecture this property for arbitrary and prove it in the limit cases and . Moreover, for partitions of a given integer , they make the stronger suggestion that the coefficients in the powers of in count certain sets of matchings i.e. fixpoint-free involutions of the symmetric group on elements (the matchings-Jack conjecture) and that the coefficients in the powers of in count certain sets of locally orientable hypermaps i.e. connected bipartite graphs embedded in a locally orientable surface (the hypermap-Jack conjecture or -conjecture). We look at the case and study a variant of the conjectures involving labelled objects defined in the following section.

In this specific case the two conjectures are related. Indeed, one has:

 Ψ(x,y,z,t,α)=αt∂∂t ∑k⩾0(−1)k+1∑μ1⋯μk∈P∖∅ ∏ipμi(y)t|μi|∑λ,ν⊢|μi|z−1λα−ℓ(λ)aλμiν(α)pλ(x)pν(z),

which implies that

 [pn(y)]Ψ(x,y,z,t,α)=αt∂∂ttn∑λ,ν⊢nz−1λα−ℓ(λ)aλnν(α)pλ(x)pν(z).

As a result, the following formula holds:

 hλnν(α)=αnz−1λα−ℓ(λ)aλnν(α). (1)

However, because of the difference in the combinatorial objects involved in the two conjectures, they are not equivalent.

###### Remark 1.

According to the definition of the coefficients , Equation (1) can be rewritten as .

### 1.2 Combinatorial background

#### Matchings

Given a non-negative integer and a set of vertices we call a matching on a set of non-adjacent edges such that all the vertices are the endpoint of one edge. Given two matchings and , the graph induced by the vertices in and the edges of is composed of cycles of even length for some and we denote . For a partition of , define two canonical matchings and . The matching is obtained by drawing a gray colored edge between vertices and for . The matching is obtained by drawing a black colored edge between vertices and for where if for some and otherwise. Obviously . Denote by the set of all the matchings on such that and . A matching in which all edges are of kind is called bipartite. This graph model is closely linked to the connection coefficients of two classical algebras.

• The class algebra is the center of the group algebra . For , denote by the formal sum of all permutations with cycle type . The set is a basis of the class algebra.

• The double coset algebra is the Hecke algebra of the Gelfand pair where is the centralizer of in . For , denote by the double coset consisting of all the permutations such that has a cycle type . The set is a basis of the double coset algebra.

We define the connection coefficients of these algebras by

 cλμν=[Cλ]CμCν and bλμν=[Kλ]KμKν,λ,μ,ν⊢n. (2)
###### Proposition 1 ([GouJac96], proposition 4.1;  [HanStaSte92], Lemma 3.2).

Using the notation above:

 bλμν/|Bn|=|Gλμν| and cλμν=|{δ∈Gλμν∣δ % is bipartite}|.

This paper is focused on the case . For we consider the set of labelled matchings , i.e. the tuples composed of a matching and a permutation on the cycles of length in for all (the cycles of length are not labelled). Clearly, .

###### Example 1.

Figure 1 depicts a labelled matching from with three labelled squares: , and .

#### Locally orientable hypermaps

One can define locally orientable hypermaps as connected bipartite graphs with black and white vertices. Each edge is composed of two half-edges both connecting the two incident vertices. This graph is embedded in a locally orientable surface such that if we cut the graph from the surface, the remaining part consists of connected components called faces or cells, each homeomorphic to an open disk. The map can also be represented (not in a unique way) as a ribbon graph on the plane keeping the incidence order of the edges around each vertex. In such a representation, two half-edges can be parallel or cross in the middle. We say that the hypermap is orientable if it is embedded in an orientable surface (sphere, torus, bretzel, …). Otherwise the hypermap is embedded in a non orientable surface (projective plane, Klein bottle, …) and is said to be non-orientable. In this paper we consider only rooted hypermaps, i.e. hypermaps with a distinguished half-edge. More details about hypermaps can be found in [lanZvo04].
The degree of a face, a white vertex or a black vertex is the number of edges incident to it. Hypermaps are also classified according to a triple of integer partitions that give respectively the degree distribution of the faces, the degree distribution of the white vertices, and the degree distribution of the black vertices. For any integer and partitions and of , denote and (resp. and ) the set and the number of locally orientable hypermaps (resp. orientable) of face degree distribution , white vertices degree distribution and black vertices degree distribution . When , the hypermap has only one white vertex. We call it a star hypermap.

###### Remark 2.

Star hypermaps are in natural bijection with unicellular hypermaps, i.e. hypermaps with only one face but an arbitrary number of white vertices. While unicellular hypermaps received a more significant attention in previous papers (see e.g. [GouSch98, MorVas13, Vas13, Vas17]), it is much more convenient to work with multicellular star hypermaps for our purpose.

###### Example 2.

Two star hypermaps are depicted on Figure 2. The leftmost (resp. rightmost) one is orientable (resp. non-orientable) and has a face degree distribution (resp. ).

###### Remark 3.

When for some integer , hypermaps reduce to classical (non-bipartite maps). The reduction is obtained by connecting the two edges incident to each black vertex and removing all the black vertices. Figure 3 gives an example of a non-bipartite star maps represented as a ribbon graph.

In this paper, we consider labelled star hypermaps, i.e. hypermaps where the black vertices are labelled by integers such that the vertex incident to the root is labelled . Denote the degree of the black vertex indexed , we further assume that the edges incident to the black vertex indexed are labelled with

 ∑1⩽j

with the additional condition that the root edge (incident to the black vertex indexed ) is labelled . For , denote the set of labelled star hypermaps with face degree distribution and black vertices degree distribution . We focus on the special case . Clearly,

 |˜Lλ[km]|=(m−1)!k!m−1(k−1)!lλn,[km]=m!k!mnlλn,[km].
###### Example 3.

The two ribbon graphs depicted on Figure 4 are labelled star hypermaps with (left-hand side) and (right-hand side).

### 1.3 Relation to the matchings-Jack and hypermap-Jack conjectures

One can show (see e.g. [GouJac96], [HanStaSte92], [Vas15a]) that the numbers , and numbers of hypermaps are linked to the coefficients and :

 aλμ,ν(1) =cλμ,ν and aλμ,ν(2)=1|Bn|bλμ,ν, hλμ,ν(1) =mλμ,ν and hλμ,ν(2)=lλμ,ν.

For general values of Goulden and Jackson conjecture the following relations between (resp. ) and sets of matchings (resp. hypermaps).

###### Conjecture 1 (Matchings-Jack conjecture, [GouJac96], conjecture 4.2).

For there exists a function such that

 aλμ,ν(β+1)=∑δ∈Gλμ,νβwtλ(δ)

and is bipartite.

###### Conjecture 2 (Hypermap-Jack conjecture, [GouJac96], conjecture 6.3).

For there exists a function such that

 hλμ,ν(β+1)=∑M∈Lλμ,νβϑ(M)

and is orientable.

## 2 Main results

We use linear operators for Jack symmetric functions to derive a new formula for the coefficients for general and which shows their polynomial properties and, as a consequence to Equation (1), the polynomial properties of the coefficients . Making this formula explicit and using some bijective constructions for labelled star hypermaps and matchings, we show a variant of the matchings-Jack and the hypermap-Jack conjectures for labelled objects in some important cases.
Denote , the Laplace-Beltrami operator. Namely,

 Dα=α2∑ix2i∂2∂x2i+∑i≠jxixjxi−xj∂∂xi

and let and be the operators on symmetric functions defined by

 Δ=[Dα,[Dα,p1/α]],Ω1=[Dα,p1/α],Ωk+1=[Δ,Ωk].

Our main result can be stated as follows

###### Theorem 1.

For any integer and , the coefficients verify:

 Autν∑λ⊢nz−1λα−ℓ(λ)aλn,ν(α)pλ=1∏i⩾1νi!(∏i⩾2Ωνi)Δν1−1(p1/α). (3)

As a consequence to Theorem 1, we have the following polynomial properties.

###### Corollary 1.

For , and are polynomials in with integer coefficients of respective degrees at most and .

Explicit computation of operators for and , allows us to show:

###### Theorem 2.

For , define If all but one part of are less or equal to , there exists a function such that

 ˜aλn,ν(β+1)=∑δ∈˜Gλνβwt(δ)

and is bipartite.

###### Theorem 3.

For and integers and with , define For all there exists a function such that

 ˜hλn,[km](β+1)=∑M∈˜Lλ[km]βϑ(M)

and is orientable.

###### Remark 4.

The focus on labelled objects and this variant of the matchings-Jack and the hypermap-Jack conjectures is motivated by the coefficients and that appear in Equation (3).

###### Remark 5.

One can notice that we consider all the partitions with any number of parts and in Theorem 2 but only the partitions of the type in Theorem 3. This is due to the existence of a distinguished (root) edge in the star hypermaps of the later theorem that prevents the extension of our methods to less symmetric cases.

## 3 Background and prior works

The following sections provide some relevant background regarding the computation of , , and , i.e. the computation of the coefficients and in the classical cases and known results for these coefficients with general .

### 3.1 Classical enumeration results for matchings and hypermaps

Except for special cases no closed formulas are known for the coefficients , , and . Prior works on the subjects are usually focused on the case . With this particular parameter, one has and . Using an inductive argument Bédard and Goupil [BedGou92] first found a formula for in the case , which was later reproved by Goulden and Jackson [GouJac92] via a bijection with a set of ordered rooted bicolored trees. Later, using characters of the symmetric group and a combinatorial development, Goupil and Schaeffer [GouSch98] derived an expression for the connection coefficients in the general case as a sum of positive terms (see Biane [Bia04] for a succinct algebraic derivation; and Poulalhon and Schaeffer [PouSch00], and Irving [Irv06] for further generalisations). Closed form formulas of the expansion of the generating series for the and and their generalisations in the monomial basis were provided by Morales and Vassilieva and Vassilieva using bijective constructions for hypermaps in [MorVas13], [Vas17] and [Vas13]. Equivalent results using purely algebraic methods are provided in [Vas15b].

### 3.2 Prior results on the Matchings-Jack conjecture and the Hypermap-Jack conjecture

While the matchings-Jack and the hypermap-Jack conjecture are still open in the general case, some special cases and weakened forms have been solved over the past decade. In particular, Brown and Jackson in [BroJac07] prove that for any partition , verifies a weaker form of the hypermaps-Jack conjecture. Later on, in his PhD thesis ([Lac09]), Lacroix defines a measure of non-orientability for hypermaps and focuses on a stronger form of the result of Brown and Jackson. He shows that

 ∑ℓ(λ)=rhλμ,[2m](β+1)=∑M∈⋃ℓ(λ)=rLλμ,[2m]βϑ(M).

In particular he proves the hypermap-Jack conjecture for . Finally, Dolega in [Dol17] shows that

 hnμ,ν(β+1)=∑M∈Lnμ,νβϑ(M)

holds true when either is restricted to the values or is general but .

Except the limit cases already covered by Goulden and Jackson [GouJac96], the matchings-Jack conjecture has be proved by Kanunnikov and Vassiliveva [KanVas16] in the case . More precisely the authors introduce a weight function for matchings in

 aλn,n(β+1)=∑δ∈Gλn,nβwtλ(δ),

besides, iff is bipartite.

In [DolFer16] and [DolFer17] Dolega and Feray focus only on the polynomiality part of the conjectures and show that the and are polynomials in with rational coefficients for arbitrary partitions . See also [Vas15a] for a proof of the polynomiality with non-negative integer coefficients of a multi-indexed variation of in some important special cases.

## 4 Proof of Theorem 1 and Corollary 1

### 4.1 Properties of Jack symmetric functions

In order to prove Theorem 1, we need to recall some known properties of Jack symmetric functions.

Pieri formulas
Given two partitions, the generalised binomial coefficients are defined through the relation

 Jαλ(1+x1,1+x2,…)Jαλ(1,1,…)=∑μ⊆λ(λμ)Jαμ(x1,x2,…)Jαμ(1,1,…).

Details about the existence and properties of these binomial coefficients can be found in [Las89, Las90]. As shown in [OkoOls97] these coefficients are equal to some properly normalised shifted Jack polynomials.
For and integer define the partition of (if it exists) obtained by replacing in by and keeping all the other parts as in . Similarly for and integer we define the partition of (if it exists) obtained by replacing in by and keeping all the other parts as in . Define also the numbers as

 ci(γ)=α(γ(i)γ)jγ(α)jγ(i)(α).

In [Las89] Lassalle showed the following Pieri formulas

 p1Jαγ =ℓ(γ)+1∑i=1ci(γ)Jαγ(i), (4) p⊥1Jαρ =α∂∂p1Jαρ=ℓ(ρ)∑i=1jρ(Dol17α)jρ(i)(α)ci(ρ(i))Jαρ(i). (5)

Power sum expansion and Laplace Beltrami operator
For , denote the coefficient of in the power sum expansion of . Namely,

 Jαλ=∑μ⊢nθλμ(α)pμ.

As shown in [Mac87] Jack symmetric functions are eigenfunctions of and verify

 DαJαλ=θλ[1|λ|−221](α)Jαλ. (6)

Furthermore, according to [KanVas16, Lemma 2], for any partition of some integer

 θγ(i)n+1(α)=θγn(α)(θγ(i)[1n−12](α)−θγ[1n−22](α)). (7)

Finally, for integers , the following relation holds ([KanVas16, Equation (30)]):

 ℓ(γ)+1∑i=1ci(γ)(θγ(i)[1n−12])a (θγ[1n−22])bJαγ(i)=Daαp1DbαJαγ (8)

Operators
Following [Las08], denote also the two conjugate operators and defined by

 E2=[Dα,p1/α]=∑i⩾1ipi+1∂∂pi, E⊥2=[p⊥1/α,Dα]=∑i⩾1(i+1)pi∂∂pi+1.

We show in [KanVas16, Theorem 5] that for and indeterminates the following relation for Jack symmetric functions holds

 ∑ρ⊢n+1θρn+1(α)Jαρ(x)E⊥2Jαρ(y)jρ(α)=∑γ⊢nθγn(α)Jαγ(y)ΔJαγ(x)jγ(α). (9)

### 4.2 Proof of Theorem 1

The first step is to show the following lemma.

###### Lemma 1.

Let and be two indeterminates. Jack symmetric functions verify

 ∑ρ⊢n+1θρn+1(α)Jαρ(x)p⊥1Jαρ(y)jρ(α)=α∑γ⊢nθγn(α)Jαγ(y)E2Jαγ(x)jγ(α). (10)
###### Proof.

Start with the second Pieri formula and then apply the known identities above. For brevity, we omit parameter in Jack symmetric functions and their coefficients in the power sum basis.

 ∑ρ⊢n+1θρn+1Jρ(x)p⊥1Jρ(y)jρ \lx@stackrel(???)=∑ρ⊢n+1ℓ(ρ)∑i=1θρn+1Jρ(x)ci(ρ(i))Jρ(i)(y)jρ(i), \lx@stackrel(???)=∑γ⊢nℓ(γ)+1∑i=1θγ(i)n+1Jγ(i)(x)ci(γ)Jγ(y)jγ, \lx@stackrel(???)=∑γ⊢nℓ(γ)+1∑i=1ci(γ)(θγ(i)[1n−12]−θγ[1n−22])θγnJγ(i)(x)Jγ(y)jγ, \lx@stackrel(???)=∑γ⊢nθγnJγ(y)(Dαp1−p1Dα)Jγ(x)jγ, \lx@stackrel(???)=α∑γ⊢nθγnJγ(y)E2Jγ(x)jγ.\qed

The key element of the proof of Theorem 1 is the following result.

###### Theorem 4.

For any integer denote the operator defined by:

 Π1=1αp⊥1=∂∂p1,Πk+1=[Πk,E⊥2].

Given two indeterminates and , the following identity holds:

 ∑ρ⊢n+kθρn+k(α)Jαρ(x)ΠkJαρ(y)jρ(α)=∑γ⊢nθγn(α)Jαγ(y)ΩkJαγ(x)jγ(α). (11)
###### Proof.

In the case Theorem 4 reduces to Equation (10). Assume the property is true for some . We have (reference to parameter is also removed)

 ∑ρ⊢n+k+1θρn+k+1Jρ(x)Πk+1Jρ(y)jρ =∑ρ⊢n+k+1θρn+k+1Jρ(x)[Πk,E⊥2]Jρ(y)jρ =Πk∑ρ⊢n+k+1θρn+k+1Jρ(x)E⊥2Jρ(y)jρ−E⊥2∑ρ⊢n+k+1θρn+k+1Jρ(x)ΠkJρ(y)jρ =Πk∑ρ⊢n+kθρn+kΔJρ(x)Jρ(y)jρ−E⊥2∑γ⊢n+1θγn+1Jγ(y)ΩkJγ(x)jγ =Δ∑ρ⊢n+kθρn+kJρ(x)ΠkJρ(y)jρ−Ωk∑γ⊢n+1θγn+1E⊥2Jγ(y)Jγ(x)jγ =Δ∑γ⊢nθγnΩkJγ(x)Jγ(y)jγ−Ωk∑γ⊢nθγnJγ(y)ΔJγ(x)jγ =∑γ⊢nθγnΔΩkJγ(x)Jγ(y)jγ−∑γ⊢nθγnJγ(y)ΩkΔJγ(x)jγ =∑γ⊢nθγn[Δ,Ωk]Jγ(x)Jγ(y)jγ.

Where the fourth and the sixth line are both obtained by applying Equation (9) and the recurrence hypothesis. As a result the property is true for . ∎

We end the proof of Theorem 1 by noticing that

 Πk=k!∂∂pk.

For an arbitrary integer partition of , rewrite Equation (11) with instead of and instead of and extract the coefficient in :

 mνp(ν)∑λ⊢nz−1λα−ℓ(λ)aλn,νpλ(x)=Ωνp∑ρ⊢n−νpz−1ρα−ℓ(ρ)νp!aρn−νp,ν∖νppρ(x). (12)

Iterating the equation above for and, then, applying times Equation (9) yields the desired formula.

### 4.3 Proof of Corollary 1

As shown e.g. by Stanley in [Sta89], the Laplace Beltrami operator can be expressed in terms of the power sum symmetric functions as:

 Dα=(α−1)2∑ii(i−1)pi∂∂pi+α2∑i,jijpi+j∂∂pi∂∂pj+12∑i,j(i+j)pipj∂∂pi+j.

Besides,

 [Dα,p1/α]=E2=∑i⩾1ipi+1∂∂pi

As a result it is clear from the definition of operators and that, for any integer partition , the coefficients in the power sum expansion of are polynomial in with (possibly negative) i