Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture

Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Do{\l}\k{e}ga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^\lambda_{\mu^0,...,\mu^k}(b)$ obtained by the expansion of these functions in the power-sum basis. These coefficients are indexed by $k+2$ integer partitions and the deformation parameter $b$, and can be considered as a generalization for $k\geq1$ of the connection coefficients introduced by Goulden and Jackson. We prove that when we take some marginal sums, these coefficients enumerate $b$-weighted $k$-tuples of matchings. This can be seen as an"unrooted"version of a recent result of Chapuy and Do{\l}\k{e}ga for constellations. For $k=1$, this gives a partial answer to Goulden and Jackson Matching-Jack conjecture. Lassale has formulated a positivity conjecture for the coefficients $\theta^{(\alpha)}_\mu(\lambda)$, defined as the coefficient of the Jack polynomial $J_\lambda^{(\alpha)}$ in the power-sum basis. We use the second main result of this paper to give a proof of this conjecture in the case of partitions $\lambda$ with rectangular shape.

1. Introduction 1.1. Jack polynomials and maps. Jack polynomials J (α) ξ are symmetric functions indexed by an integer partition ξ and a deformation parameter α that were introduced in [22]. Jack polynomials can be considered as one parameter deformation of Schur functions, which are obtained by evaluating the Jack polynomials at α = 1 and rescaling. For α = 2 we recover the zonal polynomials. This family of symmetric functions is related to various combinatorial problems [13,17,20]. Some properties of Jack polynomials have been investigated in [32] and [31,Chapter VI].
In this paper, we will be interested in relationships between Jack polynomial series and generating series of maps. A connected map is a 2-cell embedding of a connected graph into a closed surface without boundary, orientable or not. A map (1) is an unordered collection of connected maps. In this paper, we will use the word orientable for maps on orientable surfaces and the word non-oriented for maps on general surfaces, orientable or not. Maps appear in various branches of algebraic combinatorics, probability and physics. The study of maps involves various methods such as generating series, matrix integral techniques and bijective methods, see e.g. [1,2,6,14,29]. In this paper we will consider a class of vertex-colored maps that generalize bipartite maps, called k-constellations. Constellations on orientable surfaces were introduced in [29] and were generalized to the case of non-orientable surfaces in [8], see Section 1.3.
In the case k = 1, these functions were first introduced by Goulden and Jackson [17]. They suggested that the function τ (1) b is related to the generating series of matchings and Ψ (1) b is related to the generating series of connected hypermaps (or by duality connected bipartite maps). The exponent of the shifted parameter b := α − 1 is claimed to be correlated to the bipartiteness of the matchings in the first case and to the orientability of the maps in the second one. This was formulated in two conjectures that are still open, namely the b-conjecture and the Matching-Jack conjecture. These conjectures imply that the coefficients of the functions τ in powersum basis (see Equations (5) and (6)). We investigate their relationship with the enumeration of non-oriented constellations and tuples of matchings.
1.2. Main contributions. We now say a word about the main contributions of the paper. The first four points will be discussed in more details in the next subsections of the introduction: • We describe an encoding of non-oriented constellations with tuples of matchings; two versions of this correspondence are given, see Proposition 3.4 and Theorem 4.13. • This correspondence is used to obtain Theorem 1.4 which relates the generating series of non-oriented k-constellations to the function Ψ in the special case b = 1. The case k = 1 of this result was proved by Goulden and Jackson in [18]. • In the second part of this paper, we consider some marginal sums of the coefficients c λ µ 0 ,...,µ k and h λ µ 0 ,...,µ k , where we control two partitions λ and µ and the number of parts of the other partitions, denoted respectively c λ µ,l1,...l k (b) (2) The function introduced in [8] is a specialization of this function.
Algebraic Combinatorics, Vol. 5 #6 (2022) and h λ µ,l1,...,l k (b). Theorem 1.9 (see also Theorem 4.4) states that the coefficients c λ µ,l1,...l k (b) are non-negative integer polynomials in b and that they enumerate b-weighted k-tuples of matchings. The proof is based on the work of Chapuy and Dołęga [8] that gives an analogous result for the coefficients h λ µ,l1,...,l k (b). The fact that the coefficients c λ µ,l1,...l k (b) are polynomials in b with positive coefficients can directly be obtained from the result of [8], but not the integrality because of the derivative taken in Equation (2). In the proof of Theorem 1.9 we use symmetry properties to eliminate factors appearing in the denominator. When k = 1, Theorem 1.9 gives the marginal sum case in the Matching-Jack conjecture, and covers other partial results established for this conjecture [24,25]. • Theorem 5.5 gives a combinatorial expression for the coefficients of the development of Jack polynomials J λ in the power-sum basis, for rectangular partitions λ. In particular, this completes the proof of Lassalle's conjecture in the rectangular case. The proof is based on Theorem 1.9. • Theorems 6.11 and 6.12 give a combinatorial interpretation of the top degree part in coefficients c λ µ 0 ,...,µ k . In the case k = 1, this was investigated in [5] using Jack characters. We give here a different proof.

Constellations and matchings.
We consider the definition of constellations on general surfaces, orientable or not, given in [8]. The link with the usual definition of constellations in the orientable case is explained in Section 2.5.
constellation is a map, connected or not, whose vertices are colored with colors {0, 1, . . . , k} such that (3) : (1) Each vertex of color 0 (respectively k) has only neighbors of colors 1 (respectively k − 1). (2) For 0 < i < k, a vertex of color i has only neighbors of color i − 1 and i + 1, and each corner of such vertex separates two vertices of colors i − 1 and i + 1 (see Section 2.4 for the definition of corners).
Constellations come with a natural notion of rooting; a connected constellation M is rooted by distinguishing an oriented corner c of color 0. The rooted constellation obtained is denoted (M, c). We can define the size of a constellation as the sum of the degrees of all vertices of color 0. An example of a rooted non-oriented 3-constellation of size 3 is illustrated in Figure 1.
Given a k-constellation M, we can define its profile as the (k + 2)-tuple of integer partitions (λ, µ 0 , . . . , µ k ), where λ is the distribution of the face degrees, and µ i is the distribution of the vertices of color i for 0 ⩽ i ⩽ k, see Section 2.5 for a precise definition.
Constellations in the orientable case have been studied for a long time and have a well-known combinatorial description given by rotation systems; an orientable kconstellation of size n can be encoded by (k + 1)-tuple of permutations of S n . Moreover the profile of the constellation is related to the cyclic type of the permutations, see [4,7,15,29]. In Section 2.5, we introduce a notion of labelling for non-oriented constellations using right-paths. This leads to a correspondence between k-constellations and (k+2)-tuples of matchings, see Proposition 3.4. In fact, this completes the table 1.
(3) We use here the convention of [8], what we call k-constellation is often called (k + 1)constellation in the orientable case.
Algebraic Combinatorics, Vol. 5 #6 (2022) Figure 1. An example of a rooted 3-constellation drawn on the Klein bottle. The left-hand side of the square should be glued to the right-hand one (with a twist) and the top side should be glued to the bottom one (without a twist), as indicated by the white arrows. The root corner is indicated by a black arrow. Table 1 labelled bipartite maps labelled k-constellations orientable pairs of permutations [classical] (k + 1)-tuples of permutations [23] non-oriented triples of matchings [18] (k + 2)-tuples of matchings, this paper For n ⩾ 1, we will consider matchings on the set A n := {1, 1, . . . , n, n}, that is a set partition of A n into pairs. A matching δ on A n is bipartite if each one of its pairs is of the form (i, j). Given two matchings δ 1 and δ 2 , we consider the partition Λ(δ 1 , δ 2 ) of n obtained by reordering the half-sizes of the connected components of the graph formed by δ 1 and δ 2 . We consider the bipartite matching on A n given by ε := {1, 1}, {2, 2}, . . . , {n, n} , and for each λ partition of n, we set the matching The matchings ε and δ λ will have a specific role in this article as an example of bipartite matchings satisfying Λ(ε, δ λ ) = λ.
Note that the symmetric group S n is in natural bijection with the set of bipartite matchings of A n via the map σ −→ δ σ where δ σ is the bipartite matching whose pairs are (i, σ(i)) 1⩽i⩽n . Given a k-tuple of matchings, λ, µ 0 , . . . , µ k ⊢ n, we consider the two sets In Theorem 4.13, we prove that k-constellations with profile (λ, µ 0 , . . . , µ k ) with a specific labelling are in bijection with F λ µ 0 ,...,µ k .
1.4. Generating series of constellations. As described in Section 1.3, constellations on the orientable case can be encoded with tuples of permutations. The multivariate generating series of permutations with respect to their cycle type, can be related via representation theory tools to the function τ (k) 0 , see e.g. [19, Proposition 1.1] (in the following, we formulate this result using bipartite matchings rather than permutations). This gives a formula for the generating series of orientable constellations. To state this result we need to introduce the following notation; for a partition λ and a non-negative integer n we write λ ⊢ n if λ is a partition of n and if λ = [λ 1 , λ 2 , . . .] we set as in [31] where m i (λ) is the number of parts of λ equal to i. For a given set of variables u := (u 1 , u 2 , . . .), we denote u λ := u λ1 u λ2 . . ..
In this paper, we give an analogous result in the non-oriented case: is the number of parts of λ and C λ µ 0 ,...,µ k is the number of non-oriented rooted connected k-constellations with profile (λ, µ 0 , . . . , µ k ).
For k = 1, these results have been proved in [18]. In this case, constellations are bipartite maps. The generating series of non-oriented bipartite maps have been (4) investigated through the correspondence between bipartite maps and matchings on the one hand, and a relationship between matching enumeration and the structure coefficients of the double coset algebra of the Gelfand pair (S 2n , B n ) on the other hand. In this paper, we extend this development to constellations. (4) Actually, the maps considered in [18] are face colored maps called hypermaps. These maps are obtained from bipartite maps by duality.
Algebraic Combinatorics, Vol. 5 #6 (2022) 1.5. Generalized Goulden and Jackson conjectures. We define the coefficients c λ µ 0 ,...,µ k and h λ µ 0 ,...,µ k for partitions λ,µ 0 ,. . . ,µ k ⊢ n ⩾ 1 such that For k = 1, these coefficients were introduced by Goulden and Jackson. They have conjectured that these coefficients are non-negative integer polynomials in b and that they enumerate respectively matchings and bipartite maps. Based on the particular cases of Theorems 1.3 and 1.4, and on computational explorations we formulate a generalization of these conjectures for k ⩾ 1. These generalized conjectures are somehow implicit in [8]. First, we introduce a positivity conjecture. It turns out that Goulden and Jackson's positivity conjecture for the coefficients c (the case k = 1) implies the conjecture for any k ⩾ 1, since these coefficients satisfy a multiplicativity property (see Proposition 6.1). Such property does not a priori exist for the coefficients h λ µ 0 ,...,µ k , so the positivity for these coefficients is more general than the conjecture for k = 1. Using computer explorations, this conjecture has been tested when k ⩽ 5 and n ⩽ 12 − k. We attach (5) the values of the coefficients c and h for k ⩽ 5 and n ⩽ 9 − k. Given Theorems 1.3 and 1.4, Conjecture 1.5 is equivalent to the two following combinatorial conjectures: where the sum runs over rooted connected k-constellations with profile (λ, µ 0 , . . . , µ k ).
Alternatively, you can download the source files from arXiv.org Algebraic Combinatorics, Vol. 5 #6 (2022) the construction and has been open for twenty years. Dołęga and Féray have proved this polynomiality for coefficients c λ µ 0 ,µ 1 (b), see [11]. Shortly after, they deduced the polynomiality of h λ µ 0 ,µ 1 (b), see [12]. The proofs extend directly to any k ⩾ 1. Conjecture 1.6 and Conjecture 1.7 are closely related, since the functions τ (k) and Ψ (k) are related by Equation (2), and k-constellations can be encoded by matchings, see Proposition 3.4 and Theorem 4.13. However, we are not aware of any implication between them in the general case (even for k = 1). This difficulty to pass from one conjecture to another is due to the fact that we divide by z λ (1+b) ℓ(λ) in the definition of c λ µ 0 ,...,µ k and we should take the logarithm and the derivative to pass from Ψ (k) to τ (k) . Combinatorially, this can be explained by the fact that constellations appearing in the sum of the b-conjecture are rooted while there is no natural way to "root" the elements of the sets F λ µ 0 ,...,µ k appearing in the Matching-Jack conjecture. This notion of rooting will be discussed in more detail in Section 4. Nevertheless, we were able to overcome these difficulties in the case of marginal sums and deduce a result for c λ µ 0 ,...,µ k from the analog result of [8] on h λ µ 0 ,...,µ k . 1.6. The case of marginal sums. We consider the marginal sums of coefficients c λ µ,µ 1 ,...,µ k and h λ µ,µ 1 ,...,µ k , defined as follows: for all λ, µ ⊢ n ⩾ 1, and for all l 1 , . . . , l k ⩾ 1 set where ℓ(µ i ) denotes the number of parts of the partitions µ i . The main motivation for formulating the generalized versions of Goulden and Jackson conjectures introduced above is the following theorem due to Chapuy and Dołęga [8] that establishes the generalized version of b-conjecture for the marginal sums h λ µ,l1,...,l k ; As explained above, the implications between the b-conjecture and the Matching-Jack conjecture are still open problems. The proof that we give here to deduce Theorem 1.9 from Theorem 1.8 cannot be applied in the general case of the conjectures, since it uses a property of symmetry of the statistic ν that appears in Theorem 1.8, see Equation (21). Note that for the other partial results established for these conjectures (the cases b = 0, b = 1, and polynomiality), we start by proving the result for the Matching-Jack conjecture and then we deduce it for the b-conjecture, this approach is reversed in the current case.
It is known that the coefficients θ (α) µ (λ) are polynomials in α with integer coefficients (see [27] and the discussion of [11, Section 3.6]). Actually, it has been proved that these quantities are also polynomials in the multirectangular coordinates of the partition λ (see [30] for a precise definition). Lassalle has conjectured that these polynomials satisfy some positivity properties: In this paper, we consider the case where the partition λ has a rectangular shape, i.e. a partition with q parts of size r, where r, q ⩾ 1. In this case, the multirectangular coordinates of λ are given by (q, r) and we write λ = (q × r), and θ Using a recurrence formula for the coefficients θ (α) µ (q, r), Lassalle has established in the same paper the positivity in Conjecture 1.10 for the rectangular case but not the integrality. In [13], a combinatorial expression of these coefficients in terms of weighted bipartite maps was given. However the weights considered are not integral.
In Theorem 5.5, we give a complete answer to the rectangular case in Conjecture 1.10 by proving the integrality of the coefficients. We obtain an expression of (−1) |µ| z µ θ (α) µ (q, −r) as a sum of bipartite maps with monomial weights in q, −αr and b. The approach we use here is different from the one used in [30] and [13]. It is based on the marginal case in the Matching-Jack conjecture (Theorem 1.9) and Corollary 5.2 that relates the Jack polynomials indexed by rectangular partitions to some specializations of the function τ 1.8. Outline of the paper. In Section 2 we introduce some necessary definitions and notation. In Section 3, we introduce a notion of labelling for non-oriented constellations, in order to construct a correspondence between k-constellations and matchings. Building on that, we prove Theorem 1.4. Section 4 is devoted to the proof of Theorem 1.9. In Section 5, we prove the rectangular case in Conjecture 1.10. In Section 6, we discuss some general properties of the generalized connection coefficients c λ µ 0 ,...,µ k and we give a new proof for the positivity of the top degree part of these coefficients.

Preliminaries
The quantity ℓ is called the length of λ and is denoted ℓ(λ). The size of λ is the integer |λ| := λ 1 + λ 2 + · · · + λ ℓ . If n is the size of λ, we say that λ is a partition of n and we write λ ⊢ n. The integers λ 1 ,. . . ,λ ℓ are called the parts of λ. For every i ⩾ 1, we denote by m i (λ) the number of parts of λ which are equal to i. The partition 2λ is given by 2λ : Algebraic Combinatorics, Vol. 5 #6 (2022) We denote by P the set of all partitions, including the empty partition. For every partition λ and i ⩾ 1, we set λ i = 0 if i > ℓ(λ). The dominance partial ordering ⩽ on P is given by We identify a partition λ with its Young diagram defined by Fix a box □ := (i, j) ∈ λ. Its arm-length is given by a λ (□) := |{(i, r) ∈ λ, r > j}| and its leg-length is given by ℓ λ (□) := |{(r, j) ∈ λ, r > i}|. Two α-deformations of the hook-length product were introduced in [32]: With these notations, the classical hook-length product is given by (see e.g. [32])

2.2.
Matchings. We introduce some notation related to matchings as defined in [17]. We recall that for every n ⩾ 1, we set A n := {1, 1, . . . , n, n}. We also denote by F n the set of matchings on A n . For δ 1 , . . . , δ r ∈ F n we denote by G(δ 1 , . . . , δ r ) the multigraph with vertex-set A n , and edges all the pairs of δ 1 ∪ . . . ∪ δ k . In the case r = 2, we note that all connected components of G(δ 1 , δ 2 ) are cycles of even size, so we can define Λ(δ 1 , δ 2 ) as the partition of n obtained by taking half of the size of each connected component of G(δ 1 , δ 2 ).

Symmetric functions and Jack polynomials.
For the definitions and notation introduced in this subsection we refer to [31]. We denote by S the algebra of symmetric functions with coefficients in Q. For every partition λ, we denote m λ the monomial function, p λ the power-sum function and s λ the Schur function associated to λ. If α is an indeterminate, let S α := Q[α] ⊗ S denote the algebra of symmetric functions with rational coefficients in α. We recall the following notation introduced in Section 1.4; We denote by ⟨., .⟩ α the α-deformation of the Hall scalar product defined by Macdonald [31, Chapter VI.10] has proved that there exists a unique family of symmetric functions (J λ ) in S α indexed by partitions, satisfying the following properties, called Jack polynomials; where [m µ ]J λ denotes the coefficient of m µ in J λ , and 1 n is the partition with n parts equal to 1. For α = 1 and α = 2, the Jack polynomials are given by where Z λ denotes the zonal polynomial associated to λ, see [31,Chapters VI and VII]. The squared norm of Jack polynomials can be expressed in terms of the deformed hook-length products, (see [32,Theorem 5.8]): In particular, we have We conclude this subsection with the following theorem (see [31,Equation 10.25]). Another description of orientable maps is the following: a map is orientable if each one of its faces can be endowed with an orientation such that for every edge e of the map the two edge-sides of e are oriented in opposite ways. In Figure 3 we have an edge e whose sides are incident to two faces F 1 and F 2 (not necessarily distinct), and that are oriented in opposite ways. In this case we say that the orientation of the faces is consistent. A pair of edge-sides that appear consecutively while travelling along a face F is called a corner of F . An oriented corner is a corner endowed with an order on its pair of edge-sides. A corner of a vertex v is a corner whose edge-sides are incident to v. In this case, we say that the corner is incident to v. In this paper, we will consider rooted maps, i.e. maps with a distinguished oriented corner. We call canonical orientation of a rooted connected orientable map the unique orientation on the faces of the map which is consistent and such that the face containing the root is oriented by the root corner. Generating series of non-oriented constellations 2.5. k-Constellations. In this subsection, we introduce the same notation related to constellations as in [8].
We say that a corner of a constellation has color i if it is incident to a vertex of color i. We call a right-path of a k-constellation M, a path of length k along the boundary of a face of M that separates a corner of color 0 with a consecutive corner of color k incident to this face. We recall that a connected k-constellation M is rooted if it is equipped with a distinguished oriented corner c of color 0. This oriented corner c is called the root of the constellation. This is equivalent to distinguishing in M, that will be the right-path following the root corner, see Figure 4. We will use the term root to design the root corner or the root right-path. The rooted constellation will be denoted (M, c). We say that an edge is of color {i, i + 1} if its end points are of color i and i + 1. When k = 1, 1-constellations are bipartite maps and right-paths are edge-sides.
Since the number of right-paths contained in each face is even, we can define the degree of a face as half the number of its right-paths. Similarly, we define the degree of a vertex as half the number of right-paths that passes by this vertex (we can see that this is the number of edges incident to this vertex if it has color 0 or k, and half the number of edges incident to this vertex if it has color in {1, . . . , k − 1}). We also define the size of a k-constellation M as half the number of its right-paths, it will be denoted |M|. Therefore, for every k-constellation M and for every color 0 Finally, we say that a k-constellation of size n is labelled if it is equipped with a bijection between its right-paths and the set A n = {1, 1, . . . , n, n}. Labelled kconstellations will be decorated with a check, as in | M. We now compare the definition of constellations that we use here (given in Definition 1.1) to the usual description of orientable constellations given by hypermaps, (see e.g. [4,7,15]). This correspondence between the two descriptions is mentioned in [8] without details. • The degree of each hyperedge is equal to k + 1.
• The degree of each face is a multiple of k + 1.
• There exits a consistent orientation of the faces such that when we travel along a face in this orientation we read the colors {0, 1, 2 . . . , k, 0, . . .}. A connected orientable k-constellation is rooted if it has a distinguished hyperedge.
Let us prove that this definition is equivalent to the definition of constellations that we use in this paper (see Definition 1.1).
To each connected rooted orientable k-constellation (in the sense of Definition 1.1), we can associate a hypermap as follows: we travel along each face with respect to the canonical orientation (see Section 2.4), and we add an edge between each corner of color 0 and the following corner of color k. In other terms, we close each right-path traversed from its corner of color 0 to its corner of color k by adding an edge of color (0, k), thus forming a face of degree k. Such face will be considered as a hyperedge of the hypermap. The other faces of the map will be considered as faces. In Figure 6, we have an example of this transformation illustrated on a planar 2-constellation. Let us prove that the map obtained is a hypermap; since the constellation is orientable, the orientations from either side of a given edge e of the constellation are consistent. This implies that one of the two right-paths that contain e is traversed form the corner of color 0 to the corner of color k and the other right-path will be traversed in the opposite way. Only the first right-path will be transformed to a hyperedge. This proves that e separates a hyperedge and a face. Hence the map obtained is a hypermap that satisfies the properties of Definition 2.4. Moreover, this constellation can be rooted by distinguishing the hyperedge associated to the root right-path. We Algebraic Combinatorics, Vol. 5 #6 (2022) thus recover the usual definition of orientable constellations. Conversely, if we have a hypermap with the properties of Definition 2.4, we can delete the edges of color (0, k) to obtain a constellation as described in Definition 1.1.
Remark 2.5. Note that the orientability of the constellation is necessary to obtain a hypermap by the transformation described above. The description of orientable constellations with hypermaps has the advantage of being symmetric in the k + 1 colors, while in the definition with right-paths the colors 0 and k have a particular role. This lack of symmetry seems inevitable in the case of non-oriented constellations. The purpose of this subsection is to give a bijection between labelled k-constellations of size n and (k + 2)-tuples of matchings on A n . This is a generalization of the construction given in [18] which corresponds to the case k = 1.
M is a labelled k-constellation of size n, we define M( | M) as the (k + 2)-tuple (δ −1 , δ 0 , . . . , δ k ) of matchings on A n defined as follows: • δ −1 (respectively δ k ) is the matching whose pairs are the labels of right-paths of the same face, that have a corner of color 0 (respectively k) in common. • For i ∈ {0, . . . , k − 1}, δ i is the matching whose pairs are the labels of rightpaths having an edge of color (i, i + 1) in common.
It is easy to see that the profile of a k-constellation | M can be determined by the associated matchings M( | M): • Λ(δ −1 , δ k ) is the face-type.   From the definition of a map, we know that the faces of a constellation are isomorphic to open polygons. This implies every map can be obtained by gluing polygons as above (see [29,Construction 1.3.20] for a complete proof in the orientable case). We deduce the following proposition: Finally, we use the previous correspondence between constellations and matchings to introduce the notion of duality that will be useful in Section 4. Remark 3.6. It is straightforward from the definition that duality is an involution that exchanges faces with vertices of color 0, and vertices of color i with vertices of color k + 1 − i, for 1 ⩽ i ⩽ k. More precisely, given partitions λ, µ 0 , . . . , µ k duality is a bijection between k-constellations with profile (λ, µ 0 . . . , µ k ) and constellations with profile (µ 0 , λ, µ k , . . . , µ 1 ).
Remark 3.7. It is also possible, using matchings, to generalize this notion of duality in order to exchange colors in all possible ways, while controlling the profile as in the previous remark. However, these generalizations do not have a simple description in terms of maps.
3.2. The Gelfand pair (S 2n , B n ). In this subsection, we give some results that will be useful in the proof of Theorem 1.4(i). We follow the computations given in [18] when k = 1, we recall the most important steps of this proof and give a generalized version for the key lemmas. For this purpose we need to recall some results on the Gelfand pair (S 2n , B n ) (see [31, Section VII.2]). We consider S 2n as the permutation group of the set A n := {1, 1, . . . , n, n}. We define the following action of S 2n on F n , the set of matchings on A n . This action is both transitive and faithful. We define the hyperoctahedral group B n as the stabilizer subgroup of the matching ε. One has that |B n | = n!2 n . Definition 3.9. Let σ ∈ S 2n . We define the coset-type of σ as the partition of n given by Λ(ε, σ.ε).
Lemma 3.11. For each partitions λ, µ 0 , . . . , µ k ⊢ n ⩾ 1, we have Proof. Using Equation (17) we can write We use Equation (16) to extract the coefficient of K λ from the last equality to obtain a λ µ 0 ,µ 1 ,...µ k . □ When α = 2, the Jack polynomials are called zonal polynomials and denoted by Z ξ , see [31,Chapter VII]. They can be expressed in the basis of power-sum functions as follows; for every ξ ⊢ n one has We are now ready to prove Theorem 1.4.
Using Equation (18) and Lemma 3.11, this can be rewritten as Finally, we use Lemma 3.10 and Equation (13) to conclude. □ Before deducing Theorem 1.4(ii), we introduce the following notation; if M is a k-constellation with profile (λ, µ 0 , . . . , µ k ), we define the marking (6) of M as the monomial µ k . We define the marking of a labelled constellation as the marking of the underlying constellation. Theorem 1.4(ii) can be reformulated as follows: (6) What is called marking in [8] will be called marginal marking in this paper, see Section 4.1.
Proof of Theorem 1.4(ii). Theorem 1.4(i) can be rewritten as follows; On the other hand, the number of (k + 2)-tuple of matchings (δ −1 , . . . , δ k ) with profile (λ, µ 0 . . . , µ k ) is given by (2n)! n!2 n 2 n−l(λ) n! z λ |F λ µ 0 ,...,µ k |; we have (2n)! n!2 n choices for δ −1 , n! z λ 2 n−l(λ) choices for δ k and |F λ µ 0 ,...,µ k | choices for the other matchings. Using the description of labelled k-constellations with matchings (see Proposition 3.4) we obtain where the sum is taken over labelled k-constellations, connected or not. Since the marking κ(M) is multiplicative on the connected components of M, we can apply the logarithm on the last equality in order to obtain the exponential generating series of connected labelled constellations (we use here the exponential formula for labelled combinatorial classes see e.g. [16,Chapter II]). When we forget all the labels in a connected rooted constellation except for the label "1", we obtain a constellation with a marked right-path that we can consider as a rooted constellation, see Definition 1.1.
As each rooted constellation of size n can be labelled in (2n − 1)! ways, we obtain log τ where the sum runs over connected rooted constellations. We conclude the proof by applying 2t ∂ ∂t on the last equality. □

Matching-Jack conjecture for marginal sums
The purpose of this section is to give a proof for Theorem 1.9.

Generating series of non-oriented constellations
Theorem 4.1 ([8]). For every k ⩾ 1, we have where the sum is taken over rooted connected k-constellations and ν(M, c) is a nonnegative integer which is zero if and only if (M, c) is orientable.

Moreover, we say that a b-weight ρ is integral if for every map M one has that ρ(M) is a monomial in b.
With the definition above, the quantity b ν(M,c) that appears in Theorem 4.1 is an integral b-weight on connected rooted constellations. In Section 4.3, we will consider b-weights on face-labelled constellations. For every λ, µ ⊢ n and l 1 , . . . , l k ⩾ 1 we define where F λ µ,µ 1 ,...,µ k is defined in Equation (4). Theorem 1.9 can be reformulated as follows: The purpose of this section is to use the b-weight of rooted-constellations given in Theorem 4.1 in order to define a statistic ϑ on k-tuples of matchings that satisfies Theorem 4.4. We recall that in Proposition 3.4 we have established a bijection between (k + 2)-tuples of matchings and labelled k-constellations. The difficulty here is that the sums run over k-tuples of matchings (we recall that in definition of F λ µ 0 ,...,µ k we fix the matchings δ −1 to be ε and the matching δ k to be δ λ ; see Equation (4)). It turns out that the convenient objects to consider are the face-labelled constellations. The purpose of Sections 4.2, 4.3, and 4.4 is to introduce face-labelled constellations and define b-weights on them. In Section 4.6 we will establish a bijection between F λ µ 0 ,...,µ k and face-labelled constellations.

4.2.
Face-labelled constellations. Face-labelled maps were introduced in [5] in the case of bipartite maps, we give here an analog definition for constellations. We say that a k-constellation M is face-labelled if each face is rooted (with a marked oriented corner of color 0 or equivalently with a marked right-path), and for every j > 0, the faces of degree j are labelled i.e. if M contains m j > 0 faces of degree j, Algebraic Combinatorics, Vol. 5 #6 (2022) these faces are labelled by {1, 2, . . . , m j }. Face-labelled constellations will be denoted with a hat: M. In each face, the marked corner or right-path is called the face-root. We say that a connected face-labelled k-constellation is oriented if the underlying rooted constellation is orientable, and the orientations given by the face roots are consistent, see Figure 3. Finally, we say that a connected face-labelled k-constellation M is rooted if the underlying constellation has a root c such that the orientation of the root face (given by the definition of a face-labelled constellation above) is the same as the orientation induced by the root c. This constellation will be denoted ( M, c). Note that the root c of the constellation is not necessarily a face-root.   We now define b-weights for unrooted connected face-labelled constellations. These b-weights are given by different ways to root a face-labelled constellation. Note that the b-weight ⃗ ρ has the advantage of being integral, however it is a priori less symmetric than ρ. The purpose of the next subsection is to show that the b-weights ⃗ ρ, ρ and ρ are equivalent when we sum over connected face-labelled k-constellations of a given marginal profile (λ, µ, l 1 , . . . , l k ).

4.4.
Equivalence between the three b-weights. We start by the equivalence between ⃗ ρ and ρ.
where the sum is taken over face-labelled constellations that can be obtained from (M, c) by labelling its faces, with the condition that the root face is always labelled by 1 and rooted by c (see Definition 4.7 item (1)). We deduce that the left-hand side of Equation (20) side is equal to where the sum is taken over rooted connected k-constellations with profile (λ, µ 0 , . . . , µ k ) such that the root face has maximal degree λ 1 .
On the other hand, we can rewrite the right-hand side of Equation (20) (using Definition 4.7 item (2)) as follows where the sum is taken over face-labelled rooted constellation, for which the root is in a face of maximal degree. We use Equation (19) to conclude. □ Algebraic Combinatorics, Vol. 5 #6 (2022) The link between the two b-weights ρ and ρ is less obvious. We need a property of symmetry of the b-weight defined in [8] on rooted connected constellations. We start by defining for every s ⩾ 1 the series We also define the operator π that switches the variables p ↔ q and On the other hand, one has (see [8,Corollary 5.9]) where the sum is taken over rooted connected k-constellation whose root vertex has degree s. Moreover, it is straightforward from Remark 3.6 that for every k-constellation M we have where M denotes the dual constellation of M. Applying π to Equation (21), we get We deduce the following lemma. Proof. From Theorem 4.1 we have Applying π on the last equality, we get We deduce then that the coefficient of the monomial t n p λ q µ 0 u l1 where the sum is taken over connected rooted k-constellations of marginal profile (λ, µ, l 1 , . . . , l k ). On the other hand, using Equation (23) we get that this coefficient is also equal to where the sum is taken over connected rooted k-constellations of marginal profile (λ, µ, l 1 , . . . , l k ) with the condition that the root face has degree s, which finishes the proof. □ This lemma has the following interpretation: conditioning the root to be in a face of a given degree does not affect the b-weight obtained when summing over constellations of a given marginal profile. We deduce the following corollary that establishes the equivalence claimed between ρ and ρ:  The following lemma establishes the connection between the generating functions of connected and disconnected constellations. It is a variant of the exponential formula in the combinatorial class theory. However, one has to take care of the multiplicities since we have a separate labelling for each size of faces. We give here the proof in completeness.  Finally, we notice that the marking and the quantity (1 + b) ℓ(λ) are multiplicative which concludes the proof. □ 4.6. Face-labelled constellations and Matchings. Let M be a face-labelled constellation of face-type λ. We describe a canonical way to obtain a labelled constellation | M from M that will be useful in the next proposition. We start by defining the following order on A n : 1 < 1 < 2 . . . < n < n. We label the right-paths starting from faces of highest degree and smallest label: We start from the face of degree λ 1 and label 1. We travel along this face starting from the right-path preceding the root corner, and we attribute to each right-path the smallest label not yet used. We restart with the face of highest degree and smallest label whose right-paths are not yet labelled. This will be called the canonical labelling of the face-labelled constellation M. Note that for every face-labelled constellation M of face-type λ, the matchings δ −1 and δ k associated to the canonical labelling | M (as in Definition 3.1) satisfy δ −1 = ε and δ k = δ λ , where δ λ is the matching defined in Equation (3).
We now prove the following proposition that establishes a correspondence between face-labelled k-constellations and k-tuples of matchings. M as a canonical labelling. We now prove the second part of the proposition. Let M be a face-labelled constellation. We start by the following remark: when we travel along the boundary of each face of M in the orientation induced by its root, a right-path H in M is traversed from the corner of color 0 to the corner of color k (respectively from the corner of color k to the corner of color 0) if it has a label in the first class of A n (respectively in second class) with respect to the canonical labelling of M. Indeed, this property Algebraic Combinatorics, Vol. 5 #6 (2022) is clear for the root right-path of each face. Moreover, this can be extended to the other right-paths since when we travel along a face we alternate right-paths traversed form 0 to k and right-paths traversed from k to 0, and when we traverse a connected component of G(ε, δ λ ) we alternate labels of first and second class.
We recall that M is oriented if and only if the orientations induced by the faces roots are consistent as in Figure 3. Note that the orientations of the faces of M are consistent from either side of edges of color (i, i + 1) if and only if each two right-paths having an edge of color (i, i + 1) in common are traversed in opposite ways. By the previous remark, this is equivalent to saying that δ i is bipartite. In particular M is oriented if and only if each one of the matchings δ 0 ,. . . ,δ k−1 is bipartite. Definition 4.14. Let λ, µ ⊢ n and l 1 , . . . , l k ⩾ 1. For each (δ 1 , . . . , δ k ) ∈ F λ µ,l1,...,l k , we define the non-negative integer ϑ λ (δ 0 , . . . , δ k−1 ) such that where M is the face-labelled constellation associated to (δ 1 , . . . , δ k ) by the bijection of Theorem 4.13.
Since the bijection of Theorem 4.13 ensures that M is oriented if and only if the matchings δ 0 , . . . , δ k−1 are bipartite, we note that ϑ λ (δ 0 , . . . , δ k−1 ) is equal to zero if and only if each one of the matchings δ 1 ,. . . ,δ k is bipartite.
of maximal size that does not contain any of the boxes □ i , and let n be its size. Then, we have denotes the extraction symbol with respect to the variable t.
Proof. Recall that Let □ 0 be fixed box, and let u := −c α (□ 0 ) be the opposite of its α-content, see Section 2.1. Using Theorem 2.1, we can see that J ξ (u) = 0 if and only if □ 0 ∈ ξ. In particular, the partitions that contribute to the sum of Equation (26) are the partitions that do not contain anyone of the boxes □ i , for 1 ⩽ i ⩽ s. By definition, the only partition of size n that fulfills this condition is the partition λ. Hence This concludes the proof of the lemma. □ In the case of rectangular partitions, Equation (25) has a simpler expression. Proof. It is enough to prove that The last equality can be checked directly from Theorem 2.1 and Equation (9). □ The purpose of the following lemma is to explain how to add faces of degree 1 on b-weighted bipartite maps. We will need a variant of Equation (24) where we replace ⃗ ρ by another b-weight on face-labelled maps ⃗ ρ SY M that we now define. As noticed in Remark 4.3, the b-weight b ν(M,c) that we consider in Section 4 is not the only one that satisfies Theorem 4.1. We consider now the b-weight ρ SY M (M, c) defined in [8, Remark 3], which is not integral but has more symmetry properties that will be useful in the proof of Lemma 5.3. We define ⃗ ρ SY M as the b-weight on face-labelled bipartite maps obtained in Section 4.5 when we replace b ν(M,c) by ρ SY M (M, c) in Section 4 (see also Definition 4.5 and Definition 4.7 (1)). With the same arguments used in Section 4, one can check that Equation (24) also holds for ⃗ ρ SY M . Proof. We start by proving that for every partition ξ ⊢ ℓ we have the following equation: (29) 2(m 1 (ξ) + 1)[p ξ∪1 t ℓ+1 ]τ (1) b (−t, p, q, −rα) = 2(n − ℓ)[p ξ t ℓ ]τ (1) b (−t, p, q, −rα). Using Equation (24) for the b-weight ⃗ ρ SY M introduced above, we can see that the two terms of the previous equation are generating series of bipartite maps. Hence, the last equality can be rewritten as follows: where the sums run over face-labelled bipartite maps, of face-type ξ ∪ 1 in the left hand-side and ξ in the right hand-side. The factor 2 in the left hand side of the last equation will be interpreted as marking an edge-side on the face of degree 1 with the highest label of each face-labelled bipartite map M of face-type ξ ∪ 1. Such a map can be obtained by adding an edge e with a marked side to a bipartite map M of face-type ξ so that the marked side is in a face of degree 1 in the map M ∪ {e}.
In the following, we show that this corresponds to the right-hand side of Equation (30). Let M be a map of face-type ξ. We have two ways to add such an edge e with a marked side to M: • We add an isolated edge with a marked side. We chose the highest label for the face of degree 1 that we form by adding e. We thus obtain a face-labelled map. In this case we have: w (α) ( M ∪ {e}, q, r) = 2nα · w (α) ( M, q, r); the black vertex has weight q, the white −rα, and we multiply by −1 for adding an edge. Finally we have two choices for the marked edge-side. • We choose a side of an edge s to which we add the marked side of the edge e in order to form a face of degree 1. Since the map is of size ℓ we have 2ℓ choices for the edge-side s. Once s is fixed, we chose the highest label for the face of degree 1 formed by adding e. Since we have two choices of the orientation of this face, we obtain two face-labelled maps of face-type ξ ∪ 1, that we denote M 1 and M 2 . They satisfy w (α) ( M 1 , q, r) = w (α) ( M 2 , q, r) = − w (α) ( M, q, r). On the other hand, we claim that the b-weight ⃗ ρ SY M defined above has the following property: if e is an edge that we add to a bipartite map M to form a face of degree 1 then we have: • Let us explain how to obtain this property. As explained above ⃗ ρ SY M is obtained from ρ SY M by duality (see Definition 4.5 and Definition 4.7(1)). Notice that adding a face of degree 1 on a map is equivalent to adding a white vertex of degree 1 on the dual map. But such operation does not affect the b-weight ρ SY M (this is clear from the combinatorial model used in [8] and the definition of ρ SY M [8,Remark 3]). Finally, observe that when e is not an isolated edge, one of the possible orientations of the added face does not affect the b-weight ρ SY M ( M), and for the second one ρ SY M ( M) is multiplied by b (see Definition 4.5). This concludes the proof of the previous property and thus the proof of Equation (29).
Using Equation (29), we prove by induction on ℓ that [p µ∪1 ℓ−m t ℓ ]τ  This equation can be seen as a b-deformation of the first Virasoro constraint for bipartite maps (see [26] and [8,Equation (17)]). After the first version of this article has been made public, Virasoro constraints have been proved in a greater generality by Bonzom, Chapuy and Dołęga (see [3,Proposition A.1]).
We now prove the main result of this section.
Theorem 5.5. For every partition µ ⊢ m ⩾ 1 such that m 1 (µ) = 0, we have that (−1) m z µ θ (α) µ (q, r) is a polynomial in (q, −r, b) with non-negative integer coefficients. More precisely, we have where the sum is taken over face-labelled bipartite maps of face-type µ.
Let us now prove that Equation (31) implies the positivity and the integrality of the coefficients of (−1) m z µ θ (α) µ (q, r). Since (−1) m w (α) ( M, q, r) is a polynomial in (q, −r, b) with non-negative integer coefficients, it suffices to eliminate the term α ℓ(µ) that appears in the denominator of the right-hand side of Equation (31). We say that a bipartite map is weakly face-labelled if it is obtained from a face-labelled bipartite map for which we keep the labelling of faces, but we forget the orientation of all the faces except for the face of maximal degree and smallest label in each connected component. For such a map, we have a natural notion of rooting for every connected component given by the root of the face of maximal degree and minimal label. Using a variant of Equation (19), Equation (31) can be rewritten as follows (32) z µ θ (α) µ (q, r) = where the sum is taken over weakly face-labelled bipartite maps M with face-type µ and the product runs over the connected components of M rooted as explained above.
To conclude, notice that it is direct from the definition that w (α) (M i , q, r) is divisible by α. □