TOP DEGREE PART IN b-CONJECTURE FOR UNICELLULAR BIPARTITE MAPS

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x,y, z; 1, 1 + β) with an additional parameter β that may be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has the property that for β ∈ {0, 1}, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of ψ are polynomials in β with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of β is an integer-valued statistic that in some sense “measures the non-orientability” of the corresponding bipartite map. We show that except for two special values of β = 0, 1 for which the combinatorial interpretation of the coefficients of ψ is known, there exists a third special value β = −1 for which the coefficients of ψ indexed by two partitions μ, ν, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by μ/ν, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of ψ. As a consequence, we introduce a collection of integer-valued statistics of maps (η) such that the top-degree of the multivariate generating series of rooted bipartite maps with only one face (called unicellular) with respect to η gives the top-degree of the appropriate coefficients of ψ. Finally, we show that the b-conjecture holds true for all rooted, unicellular bipartite maps of genus at most 2.


INTRODUCTION
1.1. Maps. Roughly speaking, a map is a graph drawn on a certain topological surface such as the sphere or the Klein bottle (see Section 2.3 for precise definitions). This simple object carries both combinatorial and geometric informations so it turned out that maps appear naturally in many different contexts. In particular, they have deep connections with various branches of discrete mathematics, algebra, or physics (see e.g. [LZ04,Eyn16] and references therein). One of the major steps in the study of maps is developing methods for their enumeration (either by generating functions, matrix integral techniques, algebraic combinatorics, or bijective methods) which is now a well established domain on its own. Moreover, in many areas in mathematics and physics map enumeration is crucial. We refer to [Sch15] -a great introductory text on the enumeration of maps -which shows how studying enumerative properties of maps is of great importance in many different contexts.
In this paper we will focus on an interesting connection, explored mainly by Goulden and Jackson [GJ96a], between map enumeration and symmetric functions theory. This connection led to a twenty years old open problem, called b-conjecture, that will be the main subject of this paper. Figure 1. Example of a rooted, bipartite non-orientable map with the face distribution τ = (12, 4), the black vertex distribution ν = (3, 3, 2), and the white vertex distribution µ = (3, 2, 2, 1). Faces are indicated by blue and red shaded regions and the root is indicated by a thick beginning of the selected edge with a distinguished side. The map is drawn on a projective plane: the left side of the square should be glued to the right side, as well as bottom to top, as indicated by the arrows.
1.2. Enumeration of bipartite maps in terms of Jack polynomials. We define a map as a connected graph embedded into a surface (i. e. compact, connected 2-manifold without boundary) in a way that the faces (connected components of the complement of the graph) are simply connected. A hypermap (and, by duality, bipartite map) is a face two-colored map (vertex two-colored map), so that each edge separates faces (vertices) of different colors. A map is rooted by distinguishing the root, that is the unique side and the beginning of the selected edge. A rooted hypermap (bipartite map, by duality) has the black root face (vertex, respectively), by convention, where root face (vertex, respectively) is the unique face (vertex, respectively) incident to the root. An example of a bipartite map is illustrated in Fig. 1.
Let m τ µ,ν ( m τ µ,ν , respectively) be the number of rooted hypermaps on orientable (all, respectively) surfaces, such that µ lists the degrees of black faces, ν lists the degrees of white faces and τ lists the degrees divided by two of vertices (since a map is face two-colored, all vertex degrees are even numbers). By duality m τ µ,ν ( m τ µ,ν , respectively) is also the number of rooted bipartite maps on orientable (all, respectively) surfaces, such that µ lists the degrees of black vertices (we say µ is a black vertex distribution), ν lists the degrees of white vertices (ν is a white vertex distribution) and τ lists the degrees divided by two of faces (τ is a face distribution). As standard in enumerative combinatorics, we will consider the multivariate generating series (m.g.s, for short) for these objects: (1) M (x, y, z; t) = n≥1 t n |τ |=|µ|=|ν|=n m τ µ,ν p τ (x)p µ (y)p ν (z), (2) M (x, y, z; t) = n≥1 t n |τ |=|µ|=|ν|=n m τ µ,ν p τ (x)p µ (y)p ν (z), where p τ (x) is a power-sum symmetric function, i.e.
p τ (x) = i p τ i (x), p k (x) = x k 1 + x k 2 + · · · for k ≥ 1, and |µ| := µ 1 + µ 2 + · · · denotes the size of the list µ. The use of the power-sum symmetric functions as formal variables in the above m.g.s is justified by the remarkable relation, explored by Goulden and Jackson, between bipartite maps enumeration, and symmetric functions theory. Let J (α) λ (x) be the Jack symmetric function indexed by a partition λ in the infinite alphabet x and let ·, · α be the α-deformation of the Hall scalar product on the space of symmetric functions (see Section 2.2 for a precise definition). We also use Y for the set of all integer partitions. Goulden and Jackson defined in their article [GJ96a] a family of coefficients h τ µ,ν (α − 1) µ,ν,τ by the following equation: (3) ψ(x, y, z; t, α) := αt ∂ ∂ t log λ∈Y J (α) where µ, ν, τ n means that µ, ν and τ are three partitions of size n. This rather involved definition is motivated by the below described combinatorial interpretations of ψ for two particular values of α.
In other words, Theorem 1.1 says that for any partitions µ, ν, τ n the coefficient h τ µ,ν (0) = m τ µ,ν counts some rooted, orientable, bipartite maps and h τ µ,ν (1) = m τ µ,ν counts some rooted, general (i.e. orientable or not), bipartite maps. Thus we may wonder whether, for a general β := α − 1, the quantity h τ µ,ν (β) also admits a nice combinatorial description. Note that h τ µ,ν (β) is a priori a quantity depending on a parameter α, and describing it as a quantity depending on a different parameter β := α − 1 might seem be artificial. However, it turned out that this shift seems to be a right one for finding a combinatorial interpretation of h τ µ,ν (β), as suggested by Goulden and Jackson [GJ96a] in the following conjecture. Conjecture 1.2 (b-conjecture). For all partitions τ, µ, ν n ≥ 1 the quantity h τ µ,ν (β) can be expressed as: where the summation index runs over all rooted, bipartite maps M with the face distribution τ , the black vertex distribution µ, the white vertex distribution ν, and where η(M ) is a nonnegative integer which is equal to 0 if and only if M is orientable.
1.3. b-conjecture and main result. The above conjecture is still to be resolved, but some progress towards determining a suitable statistic η, based on the combinatorial interpretation of the so-called marginal sums for maps, has been made in the last two decades. Note that there is a natural bijection between the set of rooted maps with n edges, which are not necessarily bipartite, and the set of rooted, bipartite maps with the white vertex distribution given by ν = (2 n ). In particular, the following m.g.s Ψ(x, y; t, 1 + β) := n≥1 t n µ,τ 2n h τ µ,(2 n ) (β)p τ (x)p µ (y) is of special interest, as Ψ(x, y; t, 1) is the m.g.s. for rooted orientable maps, and Ψ(x, y; t, 2) is the m.g.s. for all rooted maps. A formula for Ψ(x, y; t, 1+β) involving the Selberg integral was found by Goulden, Harer and Jackson [GHJ01], who suggested that using their formula it is possible to find a combinatorial interpretation of the following marginal sum in terms of a map statistic as in Eq. (4) (here µ, τ 2n, and the summation is taken over all partitions τ which have precisely r nonnegative parts; see Section 2.1 for a precise definition). A weaker result was first established by Brown and Jackson [BJ07], who found some statistics η of maps that describe the total marginal sum A simpler description of η was found by La Croix [La 09], who used it to give a combinatorial description of l r µ (β), as suggested by Goulden, Harer and Jackson. However, not much was known about an algebraic or combinatorial structure of h τ µ,ν (β) for arbitrary partitions τ, µ, ν n, until very recently we have proved in a joint paper with Féray [DF17] the following theorem: Theorem 1.3. For all partitions τ, µ, ν n ≥ 1 the quantity h τ µ,ν (β) is a polynomial in β of degree 2 + n − (τ ) − (µ) − (ν) with rational coefficients.
In this paper we are focused on a combinatorial part of b-conjecture, especially in the case of rooted, bipartite maps, with only one face (called unicellular). Let us fix a positive integer n, and two partitions µ, ν n. According to Conjecture 1.2, there exists some statistic η on the set of all rooted bipartite maps, such that the quantity h (n) µ,ν (β) is given by the m.g.s. of rooted, unicellular, bipartite maps with the black (white, respectively) vertex distribution µ (ν, respectively). We show that except two special values of β = 0, 1 for which the combinatorial interpretation of h (n) µ,ν (β) was known, there exists a third special value β = −1 for which we provide a combinatorial interpretation of h (n) µ,ν (β). As a result we prove the following: Theorem 1.4. For all partitions µ, ν n ≥ 1 holds true for β ∈ {−1, 0, 1}, where the summation index runs over all rooted, bipartite unicellular maps M with the black vertex distribution µ, the white vertex distribution ν, and where η(M) is a nonnegative integer which is equal to 0 if and only if M is orientable.
We show that the top-degree part of the polynomial h (n) µ,ν (β) is equal, up to a sign, to its evaluation at β = −1, thus we show that it is given by some rooted, bipartite, unicellular maps with the black (white, respectively) vertex distribution µ (ν, respectively), which are called "unhandled" (the origin of this terminology will be clear later, after we define an appropriate statistic η; see Section 3). We also show that these maps are in a bijection with rooted, orientable, bipartite maps with the black (white, respectively) vertex distribution µ (ν, respectively) and with the arbitrary face distribution.
where the summation index runs over all rooted, bipartite unicellular maps M with the black vertex distribution µ, the white vertex distribution ν, and η(M) is a nonnegative integer which is equal to 0 if and only if M is orientable.

Related problems.
A second related problem is the investigation of Jack characters -suitably normalized coefficients of the power-sum symmetric function expansion of Jack polynomials. It was suggested by Lassalle that a combinatorial description of these objects might exist. This combinatorial setup was indicated by some polynomiality and positivity conjectures that he stated in a series of papers [Las08,Las09]. Although these conjectures are not fully resolved, it was proven by us together withŚniady [DFS14] that in some special cases bipartite maps together with some statistics that "measures their non-orientability" give the desired combinatorial setup. Even more,Śniady [Śni15b] found the top-degree part of the Jack character indexed by a single partition with respect to some gradation. His result states that this top-degree part can be written as a linear combination of certain functionals, where the index set is the set of rooted, orientable, bipartite maps with the arbitrary face distribution. While conjecturally it should be expressed as a linear combination of the same functionals, where the index set is a set of some special rooted, unicellular, bipartite maps. Sniady was able to find a bijection between these two index sets [Śni15a], which inspired us to investigate the combinatorial side of b-conjecture in the case of unicellular maps, presented in this paper. Note added in revision: After submission of the current paper, the aforementioned result ofŚniady appeared in a joint paper with Czyżewska-Jankowska [CJŚ17].
We cannot resist stating that there must be a deep connection between all these problems, and understanding it would be of great interest.
1.5. Organization of the paper. In Section 2 we describe all necessary definitions and background. Then, we introduce a family of statistics of the maps and we study their properties in Section 3. Section 4 is devoted to the proof of Theorem 1.4 and its consequence, which says that the family of statistics presented in the previous section describes the top-degree part of the polynomial h (n) µν (β) associated with unicellular maps. In Section 5 we introduce some special subfamily of the statistics presented in Section 3, we study their properties and we give a proof of Theorem 1.5. We finish this paper by stating some concluding remarks and questions in Section 6.
2. PRELIMINARIES 2.1. Partitions. We call λ := (λ 1 , λ 2 , . . . , λ l ) a partition of n if it is a weakly decreasing sequence of positive integers such that λ 1 + λ 2 + · · · + λ l = n. Then n is called the size of λ while l is its length. As usual we use the notation λ n, or |λ| = n, and (λ) = l. We denote the set of partitions of n by Y n and we define a partial order on Y n , called the dominance order, in the following way: Given two partitions λ ∈ Y n and µ ∈ Y m we can construct a new partition λ ∪ µ ∈ Y n+m obtained by merging parts of λ and µ and ordering them in a decreasing fashion.
2.2. Jack polynomials. In this section we recall the definition of Jack polynomials and present several known results about them. Since they are well-established (mostly in a seminal work of Stanley [Sta89]), we do not give any proof, but explicit references.
Consider the vector space Sym of the symmetric functions Λ over the field of rational functions Q(α) and endow it with a scalar product ·, · α defined on the power-sum symmetric functions basis by the following formula (and then extended by bilinearity): Here, m i (λ) denotes the number of parts of λ equal to i. This is a classical deformation of the Hall scalar-product (which corresponds to α = 1). Now, Jack polynomials J (α) λ are symmetric functions with an additional parameter α uniquely determined (see [Mac95, Section VI,10]) by the following conditions: where m λ denotes the monomial symmetric function.
2.2.1. Basic properties. We present here several well-known identities for Jack polynomials that will be useful for us later.
J (α) λ ((t, 0, 0, . . . )) = (1 + α)(1 + 2α) · · · (1 + (n − 1)α)t n for λ = (n), 0 for (λ) > 1.  2.3. Surfaces, graphs, and maps. A map is an embedding of a connected graph G into a surface S (i.e. compact, connected, 2-dimensional manifold) in a way that the connected components of S \ G, called faces, are simply connected. Our graphs may have loops and multiple edges. Maps are always considered up to homeomorphisms. A map is unicellular if it has a single face. Unicellular maps are also called one-face maps. We will call a map orientable if the underlying surface is orientable; otherwise we will call it non-orientable. In this paper we will be mostly focused on non-orientable maps.
For the purposes of the current paper it is sometimes convenient to represent a map as a ribbon graph as follows: each vertex is represented as a small disc and each edge is represented by a thin strip connecting two discs in a way that a walk along the boundary of the ribbons corresponds to the walk along the boundary of the faces of a given map. To draw such a picture it is helpful to do it in the following steps: we first draw vertices (as small discs) together with thin strips attached around them, which represent associated half-edges. By definition half-edges are obtained by removing middle-points of all the edges (each edge consists of exactly two half-edges). We call this data the stars of the map M and we denote it by S(M ). See Fig. 2(a) for an example of S(M ) associated with the map M from Fig. 1. Next, for each edge e of M , we can connect the borders of the strips representing two halfedges belonging to e in two possible ways (see Fig. 5). We connect them in a way so that after connecting all the strips from S(M ) the walk along the boundary of the ribbons corresponds to the walk along the boundary of the faces of a given map. Fig. 2(b) presents the map M from Fig. 1 represented as a ribbon graph.
A map is rooted if it is equipped with a distinguished half-edge called the root, together with a distinguished side of this half-edge. The vertex incident to the root is called the root vertex, and the edge containing the root is called the root edge. There is an equivalent way to root a map by choosing a corner (called the root corner) and its orientation, where a corner in a map is an angular sector determined by a vertex, and two half-edges which are consecutive around it. One can then define the root half-edge as the one lying to the left of the root corner (viewed from the root vertex and according to the root corner orientation). In this paper we will use both conventions depending on the situation, and we will represent rooted maps by shading the root corner and/or by indicating the root that is incident to it.
The degree of a vertex is the number of incident half-edges, or equivalently the number of incident corners, while the degree of a face is the number of corners lying in that face, or equivalently the number of edges incident to it, with the convention that an edge incident to the same face on both sides counts for two.
If M is a map, we let V (M ), E(M ) and F (M ) be its sets of vertices, edges and faces. Their cardinalities v(M ), e(M ) and f (M ) satisfy the Euler formula: where g(M ) denotes the genus of a map M , that is the genus of the underlying surface. We recall that the Euler characteristic of the surface is 2 − 2g(M ), thus g(M ) is a nonnegative integer when M is orientable or half-integer when M is non-orientable. We also denote by C(M ) the set whose elements are indexed by faces of M . For a fixed face f ∈ F (M ) the associated element in C(M ) is the set of all corners belonging to f .
A map is bipartite if its vertices can be colored in two colors in such a way that adjacent vertices have different colors (say black and white). For a rooted, bipartite map, the color of its root vertex is always taken to be black, by convention.
Let µ, ν, τ be integer partitions. We say that a rooted bipartite map M has type (µ, ν; τ ) if µ lists the degrees of black vertices (we say µ is the black vertex distribution), ν lists the degrees of white vertices (ν is the white vertex distribution) and τ lists the degrees of faces divided by two (τ is the face distribution). We denote the set of rooted bipartite maps of type (µ, ν; τ ) on orientable (all, respectively) surfaces by M τ µ,ν ( M τ µ,ν , respectively). Note that all three partitions µ, ν, τ have necessarily the same size n, which is equal to the number of edges of the corresponding map, while its lengths correspond to the number of black and white vertices and the number of faces, respectively.
From now on, all the maps are rooted, and bipartite, thus by saying a "map", what we really mean is a "rooted, bipartite map".

MEASURE OF NON-ORIENTABILITY IN b-CONJECTURE
In this section we are going to construct a function that associates with a map M a nonnegative integer η(M ) which, in some sense, measures its non-orientability. This "measure of non-orientability" gives, in some special cases, a correct answer to b-conjecture, i.e. Eq. (4) holds true.
The construction presented in this section is due to La Croix [La 09] who used it to prove that the following marginal sum can be expressed in the same form as the right hand side of Eq. (4), where the maps in the summation index are not necessarily bipartite, have r faces and the vertex distribution µ. The construction of La Croix was originally defined for all (not necessarily bipartite) maps, but in this paper we are dealing with the case of bipartite maps, thus in the following all maps will be rooted and bipartite. Moreover, at most one of M and M is orientable, and its measure of nonorientability is equal to 0 while a measure of non-orientability of the other (nonorientable) map is equal to 1. Remark. Note that there are many function η satisfying all the conditions given by Definition 3.1. Thus, the above definition gives a whole class of functions, and any such function is called a measure of non-orientability.
Let M be a rooted map. We label all its edges according to their appearance in the rootdeletion procedure. That is, the root edge of M has label 1, the root edge of M \ {e} has label 2, etc. Here, there must be a convention chosen in which connected component should be treated first, after removing a bridge. Our convention is that we first decompose the connected component with the root corner that contained the root corner of the previous map. That is, we first decompose the map M 1 from Fig. 3.
From now on, we are going to use the following notation: for the rooted map M , and for any 1 ≤ i ≤ e(M ) we denote by e i ( For a given positive integer n and partitions µ, ν, τ n we can decompose the set M τ µ,ν of maps of type (µ, ν; τ ) in the following manner: where M τ µ,ν;i is the set of maps of type (µ, ν; τ ) such that exactly i handles appeared during their root-deletion process. In other words, it is the set of rooted maps M of type (µ, ν; τ ) such that for all natural numbers k ∈ N, except i, the root of M k is not a handle. We call maps from the set M τ µ,ν;0 unhandled. Finally, we denote the finite set {1, 2, . . . , n} by [n]. Here, we present a classical, but important for us, relation between a genus of a given map and its root-deletion procedure.
Lemma 3.2. Let M ∈ M τ µ,ν;i be a rooted map with n edges, and let j(M ) denotes the number of positive integers j ∈ [n] for which the root of M j is twisted. Then the following equality holds true: where g(M ) is a genus of the map M .
Proof. We are going to prove that Eq. (11) holds true for all maps by an induction on the number of edges of M . It is straightforward to check that there is only one rooted, bipartite map with one edge. Its root edge is a bridge and it is planar (i.e. its genus is equal to 0). Thus, Eq. (11) holds true in this case. Now, we fix n ≥ 2 and we assume that Eq. (11) holds true for all maps with at most n − 1 edges. Let M ∈ M τ µ,ν;i be a map with n edges and let i(M ) denotes the number of handles appearing in the root-deletion process of M , i.e. i(M ) = i. We are going to analyze how the Euler characteristic varies during the root-deletion process. It is straightforward from the classification of root edges and from Eq. (9) that we have following possibilities: • e is a bridge.
Thus, by an inductive hypothesis Since these are all possible cases, we proved by induction that Eq. (11) holds true for any rooted, bipartite map M , which finishes the proof.
Corollary 3.3. Let M ∈ M τ µ,ν;i be a rooted map with n edges. Then Proof. The Euler formula given by Eq. (9) yields Combining it with Eq. (11) we have the following formula: It is now enough to notice an obvious inequality which comes strictly from the definition Definition 3.1 of η: which finishes the proof.
3.2. Twist involution. Let M be a rooted map such that its root edge e is a handle. We recall that there exists the unique rooted map M different from M with the root edge e which is a handle, too, and such that the half-edges belonging to e are lying in the same corners of M \ {e} = M \ {e }, as the half-edges belonging to e . Notice that the map M is, roughly speaking, obtained from M by "twisting" its root. In this section we are going to formalize and generalize the concept of "twisting edges". Remark. Note that M and τ i M are the same graphs (thus the sets E(M ), and E(τ i M ) are the same, and it makes sense to compare properties of an edge e in M to its properties in τ i M ), but it is not true in general that τ i τ j M is the same map as τ j τ i M . The following proposition resolves when the twisting operators commute.
Proposition 3.5. Fix a positive integer n, partitions µ, ν n, and a map M with n edges. Then (1) for any i ∈ [n] the operator τ i is an involution on the set of maps with black (white, respectively) vertex distribution µ (ν, respectively), (2) let I = {i 1 < · · · < i k } be a non-empty subset of [n] such that for all i ∈ I the root edge of M i is not a bridge. Then, for any permutation σ ∈ S I the map τ σ(i 1 ) · · · τ σ(i k ) M is the same, the labels of the edges in M and in τ σ(i 1 ) · · · τ σ(i k ) M coincide, and for any j ∈ [n] the root edge of M j is a bridge iff the root edge of (τ i 1 · · · τ i k M ) j is a bridge.
Proof. Let us fix i ∈ [n]. It is clear from the construction that the operator τ i preserves white and black vertex distributions. Now, notice that twisting an i-th edge in any map M does not change the labels of the first i edges and it may change the labels of the other edges only if the edge e i (M ) is a bridge in M i . Thus, the first i labels of the edges in both maps M and τ i M are the same, so τ 2 i M = M . We are going to prove the second item by an induction on size of the set I. We have already proved above that the inductive assertion holds true for |I| = 1, so let us fix an integer 1 < k ≤ n. We assume that the inductive assertion holds true for all subsets I ⊂ [n] of size smaller then k. Let I = {i 1 < · · · < i k } be a non-empty subset of [n] such that for all i ∈ I the root edge of M i is not a bridge and let σ ∈ S I . Then, there exists an integer l ∈ [k] and a permutation π ∈ S [k]\{l} such that where we use a standard notation that the word a 1 · · · a i−1 a i a i+1 · · · a n is obtained from the word a 1 · · · a i−1 a i a i+1 · · · a n by removing the letter a i . If l = 1, then the labels in M and is not a bridge since the root edge of M i 1 is not a bridge, by the inductive assertion. Thus, the labels in M , and in τ σ(i 1 ) · · · τ σ(i k ) M coincide, too. If l > 1, then the inductive assertion says that the labels of M and τ j 1 · · · τ jm M are the same for all subsets {j 1 , . . . , j m } ⊂ I of size m < k, so by the inductive hypothesis. Moreover, again by the inductive hypothesis, the labels in the maps were not changed when we were swapped operators τ with different indices. Finally, for any j ∈ [n], the graphs M j and (τ i 1 · · · τ i k M ) j are the same. Since being a bridge is the same as being a disconnecting edge of the graph, the proof is finished.
Lemma 3.6. Let η be a measure of non-orientability. Then, for all positive integers n, i and partitions µ, ν n, there exists an involution σ η on the set M (n) µ,ν;i , which has the property that Proof. We are going to construct σ η by induction on n. For n = 1 all rooted bipartite maps with n edges are unhandled so we set σ η as an empty map.
We denote by i(M ) the number of handles appearing in the root-deletion process of M . We fix n ≥ 1 and we assume that the involution σ η is already defined for all unicellular maps with at most n edges which are not unhandled. Let M be a unicellular map with n + 1 edges. Since M has a unique face, e 1 (M ) cannot be a border so there are the following possibilities: • e 1 (M ) is a handle. In this case we set σ η (M ) = τ 1 M . Clearly (−1) η(M ) = (−1) η(ση(M ))+1 and i(M ) = i(σ η (M )); • e 1 (M ) is twisted. In this case M 2 is a unicellular map with n edges. Thus, by the inductive hypothesis there exist natural numbers Let us consider two rooted maps M 1 = τ i k +1 · · · τ i 1 +1 M and M 2 = τ i k +1 · · · τ i 1 +1 τ 1 M . Both maps M 1 and M 2 have a property that after deletion of its root we obtain the same rooted map M 2 1 = M 2 2 = τ i k · · · τ i 1 M 2 (Proposition 3.5 asserts that the labels in M 2 1 and in M 2 2 , respectively, correspond to the labels of M 1 and M 2 , respectively, shifted by 1). Thus, exactly one map from M 1 and M 2 is a map M with the unique face (and its root is twisted) while the second one has two faces (and its root is a border) and we set σ η (M ) = M . Strictly from the construction, one has In this case M 2 = M 1 ∪ M 2 is a disjoint sum of two unicellular maps (and we recall the convention for labeling: the edge e 2 (M ) belongs to M 1 ). If M 1 is not unhandled, then there exists a positive integer k and natural numbers 1 ≤ i 1 < · · · < i k ≤ e(M 1 ) such that i(τ i k · · · τ i 1 M 1 ) = i(M 1 ) and such that In this case we set σ η (M ) := τ i k +1 · · · τ i 1 +1 M and since σ η (M ) 2 = σ η (M 1 ) ∪ M 2 it is clear that (−1) η(ση(M )) = (−1) η(M )+1 and If the map M 1 is unhandled, then the map M 2 is not unhandled and there exist a positive integer k and natural numbers 1 ≤ i 1 < · · · < i k ≤ e(M 2 ) such that i(τ i k · · · τ i 1 M 1 ) = i(M 1 ) and such that In this case we set σ η (M ) : Now it is straightforward from the construction and from the inductive hypothesis that if σ η (M ) associated with the rooted map M is of the form τ i k · · · τ i 1 M , then σ η (σ η (M )) is of the same form, i.e. σ η (σ η (M )) = τ i k · · · τ i 1 σ η (M ). But for each j ∈ [k] the root edge of M i j is not a bridge. Thus where the last equalities come from Proposition 3.5, which finishes the proof.
3.3. Algebraic properties of a measure of non-orientability. Let η be a measure of nonorientability and let µ, ν, τ be partitions of a positive integer n. We define the following statistic associated with η: The main purpose of this section is to investigate algebraic properties of (H η ) τ µ,ν that will be of the great importance in the proof of Theorem 1.4. From now on, we fix a positive integer n, partitions µ, ν, τ n, and a measure of non-orientability η.
Proof. Strictly from the definition of (H η ) τ µ,ν (β) given by Eq. (13), one has the following formula: where the last equality is simply a definition of a τ µ,ν;i (β) given by Eq. (14). Proposition 3.9. For any positive integer i ≥ 1 one has Lemma 3.6 says that for each i ≥ 1 there exists an involution σ η on the set M Thus, (a η ) (n) µ,ν;i (−1) = 0 which finishes the proof.
Corollary 3.10. The following equality holds true: Proof. It is enough to plug β = −1 into Eq. (15) to obtain where g := n + 1 − (µ) − (ν) and the last equality is a consequence of Proposition 3.9. An equality is obvious from Corollary 3.3.

b-CONJECTURE FOR UNICELLULAR MAPS AND MEASURE OF NON-ORIENTABILITY
4.1. Marginal sum. We are going to prove that fixing white and black vertex distributions and allowing any face distribution, the corresponding sum of coefficients in ψ(x, y, z; t, 1 + β) is given by a measure of non-orientability of the appropriate maps. The developments in this section are similar to that of [BJ07, Section 3.5] except that here we work in a more general setup (in [BJ07, Section 3.5] ν = (2 n/2 ) with n even, while here ν is an arbitrary partition) and with a slightly different function η. We start with the following proposition.
Proposition 4.1. For any positive integer n and for any partitions µ, ν n, the following identity holds true: Proof. We know that because of the trivial identity .
Using Eq. (8) and replacing the scalar product by its expression given in Eq. (6), we obtain The formula for Jack polynomials indexed by one-part partitions given in Eq. (7) leads to the following equality: But the last expression is simply equal to which finishes the proof.
We can prove now that the following marginal sum is given by a measure of non-orientability: Theorem 4.2. For any measure of non-orientability η, for any positive integer n, and for any partitions µ, ν n, the following identity holds true: where (H η ) τ µ,ν (β) are given by Eq. (13). Proof. We recall that (H η ) τ µ,ν (β) is defined as a weighted sum of some rooted, bipartite maps (see Eq. (13)). Thus, one can define a statistic (H η ) µ,ν;i (β) as the right hand side of (18) with a summation restricted to the maps with the root vertex of degree i. We are going to prove a stronger result, namely The third equality uses the combinatorial interpretation of h τ µ,ν (0) (see Theorem 1.1) while the last equality comes from Proposition 4.1. In this way we have shown that Eq. (19) implies Eq. (18). Thus, it is sufficient to prove Eq. (19).
Before we start a proof we introduce some notation. Let r 1 , . . . , r k be some positive integers such that r 1 + · · · + r k = n. Let us fix a partition µ n. We define Sp (r 1 ,...,r k ) (µ) as the set of sequences of partitions (µ 1 , . . . , µ k ) such that µ 1 r 1 , . . . , µ k r k and such that their sum gives the fixed partition µ. That is, 1≤i≤k µ i = µ. Moreover, for any positive integer i ≥ 1 and partition µ containing a part equal to i, we set We are going to prove Eq. (19) by induction on n. Let n = 1; there exists only one partition of size n. That is, µ = ν = (1). Moreover, there is only one map with one edge, and it is planar, so clearly H (1),(1);1 (β) = 1 =H (1),(1);1 . Let us fix n ≥ 2 and assume now that the inductive assertion holds true for all partitions of size smaller than n and all integers i ≥ 1. Let us fix two partitions µ, ν n and an integer i ≥ 1. Let M be a map with the root vertex of degree i, and the black (white, respectively) vertex distribution µ (ν, respectively). We are going to understand the structure of M \ {e}, where e is the root edge of M . There are two possibilities: • M \ {e} is a disjoint sum of two maps M 1 and M 2 (they are ordered, i.e. their indices matter) with root vertices of degrees i − 1, and j − 1, respectively, the black vertex distributions µ 1 , and µ 2 , respectively, and the white vertex distributions ν 1 , and ν 2 , respectively, where (µ 1 , µ 2 ) ∈ Sp (l,n−l−1) (µ ↓(i) ), (ν 1 , ν 2 ) ∈ Sp (l,n−l−1) (ν ↓(j) ), and 1 ≤ j, l + 1 ≤ n are some integers; • M \ {e} is a single map M with the root vertex of degree i − 1, the black vertex distribution µ ↓(i) , and the white vertex distribution ν ↓(j) , where 2 ≤ j ≤ n is some integer.
To finish the proof, it is enough to notice that the following recursion holds true: Above relation comes from the analysis of the process of removing the root edge from an orientable map with the root vertex of degree i, the black vertex distribution µ, and the white vertex distribution ν. Such analysis is almost identical to the analysis we did in the general case, and we leave it as an easy exercise. 4.2. Some consequences of the polynomiality and the marginal sum results. We start with an observation that polynomials h τ µ,ν (β) have some specific form: Lemma 4.3. For any positive integer n and any partitions µ, ν, τ n one has the following expansion: h τ µ,ν (β) = 0≤i≤[g/2] a τ µ,ν;i β g−2i (β + 1) i for (µ) + (ν) + (τ ) ≤ 2 + n, 0 otherwise ; where g := 2 + n − ( (µ) + (ν) + (τ )) and a τ µ,ν;i ∈ Q. Proof. La Croix proved in [La 09, Corollary 5.22] that Lemma 4.3 holds true, assuming polynomiality of h τ µ,ν (β) (that he was not able to prove). Theorem 1.3 completes the proof.
We are now ready to prove Theorem 1.4 Proof of Theorem 1.4. Theorem 1.1 already says that the cases β = 0, 1 correspond to counting maps on orientable, and all (orientable or non-orientable), respectively, surfaces. Thus, we need to establish the remaining identity: . Note now that for fixed partitions µ, ν n the variable g := 2 + n − ( (µ) + (ν) + (τ )) taken over all partitions τ n realizes a maximum for τ = (n). Hence by Lemma 4.3 and, similarly, Remark. The equality µ,ν (β) is enumerated by unhandled maps of type (µ, ν; (n)), but it is also enumerated by orientable maps with the black (white, respectively) vertex distribution µ (ν, respectively) and the arbitrary face degree. In fact, one can use the proof of Proposition 4.1 to construct a bijection between these two sets recursively. An interesting result ofŚniady [Śni15b, Corollary 0.5] states that the top-degree part of Jack characters indexed by a onepart partition can be also expressed as a linear combination of certain functions indexed by orientable maps.Śniady informed us in private communication [Śni15a] that he can construct a similar bijection, but for different "measure of non-orientability", which inspired us to initiate a research presented in this section. The connection between striking similarities in both results seems to be far from being understood.

LOW GENERA CASES AND ORIENTATIONS
In this section we are going to prove Theorem 1.5. In fact, we are going to show that there is an infinite family of measures of non-orientability for which, in low genera cases, b-conjecture holds true. We are going to define a function η O associated with O that takes as values maps and returns a nonnegative integer that, in some sense, "measures non-orientability" of the given map. This function will be defined recursively using the same procedure of deleting the root edge from the given map, as in Definition 3.1. Remark. Note that we use two different ways to make a distinction between edges: the first way is by determining their type which can be a bridge, a border, a twisted edge, or a handle. The second way is by determining their kind which can be the first or the second. Each type of edges has uniquely determined kind except a handle which can be both of the first and of the second kind. With this notation, invariant η O (M ) associated with the rooted map M is equal to the number of edges of the second kind appearing in its root-deletion process.
We are ready to restate Theorem 1.5 in the more general form: Theorem 5.2. For a set O of orientations, for any positive integer n, and for any partitions µ, ν n such that (µ) + (ν) ≥ n − 3 the following equality holds true: The next subsections are devoted to its proof.
5.2. Unicellular maps of low genera with two handles. In this section we are going to analyze the structure of the unicellular maps of low genera with two handles, which is necessary for the proof of Theorem 5.2. Since this section is very technical, we would like to precede it by a description of the main idea of the proof of Theorem 5.2. We should try to keep it light and a bit informal to motivate the reader to understand all the technicalities that will appear after this introduction and that are necessary to present the formal proof of Theorem 5.2. It is an easy exercise (later on it will be explained in details) that having the above mentioned involution we can use a simple polynomial interpolation argument to prove Theorem 5.2, thus in the following we are going to focus on the construction of such involution. Firstly, we need to understand how a map M ∈ M (n) µ,ν;2 can look like. This map has a unique face, genus 2, and exactly 2 handles appear during its root-deletion process that correspond to edges e i (M ) and e j (M ) with labels j > i. This suggests that the map σ O (M ) might be constructed by twisting some of the edges of M in the appropriate way, similar to how it was done in Lemma 3.6. However, it turned out that in some cases it is impossible and one needs a deeper analysis of the structure of the map M . In the following section, we present an example where twisting some edges never works, and we show on this specific example how to overcome this problem. We believe that this example is the accurate toy-example of the general case. 1 τ 2 2 τ 3 M does not satisfy above required properties. Indeed, the root edge of the map (τ 1 1 τ 2 M ) 2 = τ 2 M 2 is twisted. See Fig. 6(b), which is a problem since we want this root edge to be a border. If we twist it, that is we consider the map (τ 1 1 τ 2 τ 3 M ) 2 = τ 1 τ 2 M 2 , then its root edge is a border, so we fix previous problem. However, we encounter another case: half-edges h 1 (τ 1 1 τ 2 τ 3 M ) and h 1 (τ 1 1 τ 2 τ 3 M ) lie in the same face of τ 1 τ 2 M 2 , thus the root edge of τ 1 1 τ 2 τ 3 M cannot be a handles, see Fig. 6(b). A crucial observation which helps to overcome the problem is the following: the set of corners lying in the root face of τ 1 τ 2 M 2 differs from the set of corners lying in the root face of M 2 (compare Fig. 6(c) to Fig. 7(a)). We would like to fix this. One can show that if we erase the edges e 1 (M 2 ) and e 2 (M 2 ) from the map M 2 , but we do not erase the corresponding half-edges h 1 (M 2 ), h 1 (M 2 ) , h 2 (M 2 ), h 2 (M 2 ) , then there is a unique way to choose two pairs from these four half-edges and draw two new edges connecting half-edges in each pair to obtain a map M different from M 2 such that: • its root edge is a border, • the set of corners lying in its root face is exactly the same as the set of corners lying in the root face of M 2 .
This map is shown in Fig. 7(b). Moreover, the root edge of (M ) 2 is a handle of the different kind than the root edge of M 3 . Now, we can finish the construction of σ O (M ). The set of corners lying in the root face of M is the same as the set of corners lying in the root face of M 2 . Thus, if we connect the corners of M corresponding to the root edge of M by a new edge we will always create a handle (of two possible kinds). Thus, it is enough to connect these corners by a handle of different kind than the root edge of M to construct σ O (M ). In our case it is a handle of the second kind and this map is shown in Fig. 7(c). It is straightforward from this construction that if we repeat this recipe to construct σ O (σ O (M )), we construct exactly the map M . It turned out that the general situation is basically the same. The next section is devoted to the description of the structure of a map M in a general case when we cannot construct σ O (M ) simply by twisting some edges of M . This description, which is given in Lemma 5.3 and its proof are very technical. However, the reader should think that a general picture looks almost the same as the picture from this section: a map M might have a lot of edges, but only three edges play important role in the construction of σ O (M ) and their role is the same as the role of e 1 (M ), e 2 (M ), and e 3 (M ) in the above example. • for all positive integers i < k < j the root edge of M k is a bridge; • there exists a unique positive integer i < k < j such that the root edge of M k is a border. Then there are still two possible cases: . Note that the map M presented in Section 5.2.2 was exactly the second case of the second case in the above analysis. We start with a technical lemma that treats this case in general. All the symbols n, i, j, µ, ν are as above and we recall that for any m ∈ In order to prove ( ), we need to analyze the structure of the map M k+1 . This analysis will be also crucial in proving (P2), (the structure of the map M k+1 is shown on Fig. 8). ) which is also unicellular but has genus equal to 1. Indeed, it follows immediately from Euler formula Eq. (9) and from the definition of types of edges given in Section 3.1. Since the root edges of all the maps M m are bridges for k < m < j, the map M k+1 is obtained from the unicellular map M j by planting some trees into some corners of it. In particular, the edge containing h j (M ) has the same type in both maps M j and M k . Thus, it is a handle. We conclude that M k+1 \ {e j (M )} is a map with two faces -in particular it is connected which proves our claim ( ).
Above analysis says that if one removes the edge containing h j (M ) from the map M k+1 but does not remove the corresponding half-edges h j (M ) and h j (M ) , the resulting object is a map with two faces f 1 , and f 2 and with two additional half-edges h j (M ), h j (M ) such that h j (M ) lies in some corner belonging to f 1 , and h j (M ) lies in some corner belonging to f 2 (see Fig. 8). Let l 1 (l 2 , respectively) be the word given by reading consecutive corners of the face f 1 with additional half-edge h j (M ) (f 2 with additional half-edge h j (M ) , respectively) in a way that the word given by reading consecutive corners of the unique face of the map M k+1 with respect to the root orientation is given by the concatenation l 1 · l 2 of the words l 1 and l 2 . See Fig. 8. For a given word w, we denote by ← − w the word obtained from w by reading it backwards (from right to left). Then the word given by reading consecutive corners of the unique face of (τ j M ) k+1 = τ j−k M k+1 with respect to the root orientation is given by the concatenation l 1 · ← − l 2 . Now, we recall our assumption that C(M k ) = C((τ j M ) k ). This is equivalent to saying that the set of corners belonging to the root face of M k is different than the set of corners belonging to the root face of (τ j M ) k (since both M k and (τ j M ) k have exactly two faces). Thus, the half-edge h k (M ) is lying in some corner belonging to l 1 and incident to h k+1 (M ) and h k (M ) divides this corner into two new corners. Similarly h k (M ) is lying in some corner belonging to l 2 and divides it into two new corners (indeed, if both half-edges h k (M ), and h k (M ) are lying in l 1 , then, clearly, the sets of corners belonging to the root face of M k and to the root face of (τ j M ) k coincide, which gives a contradiction with our assumption). Let l 1 , l 1 and l 2 , l 2 , respectively be two new words obtained by reading consecutive corners between h k (M ) and h j (M ) and between h k (M ) and h j (M ) , respectively, as depicted in Fig. 9(a). In other words, l 1 is a concatenation of (l 1 ) − , a letter c 1 which corresponds to the corner of M k+1 containing h k (M ), and − (l 1 ), where w − ( − w, respectively) is a word obtained from w by removing its last (first, respectively) letter. Similarly, l 2 is a concatenation of (l 2 ) − , a letter c 2 which correspond to the corner of M k+1 containing h k (M ) , and − (l 2 ). Then the word given by reading consecutive corners of the root face of M k , starting from the first corner visited after the root corner, is given by the concatenation l 2 · l 1 , while the word obtained by reading the corners visited consecutively in the root face of (τ j M ) k , starting from the first corner visited after the root corner, is given by the concatenation l 2 · ← − l 1 · ← − l 2 ·l 1 . In particular, the map (τ j M ) k is unicellular, and its root is twisted, which shows that C(M k ) = C((τ j M ) k ), as we assumed.  Fig. 9(a) to Fig. 9(b)). By Definition 5.1 µ,ν;2 , let i < j be the labels of the edges that are handles in the root-deletion process of M , and let k < j be the largest positive integer such that C(M k ) = C((τ j M ) k ) (we assume that it exists). Then necessarily i < k < j and the root edge of M k is a border.
Proof. We recall from the beginning of Section 5.2.3 that there are two possible cases: and such that k is the greatest possible, then necessarily i < k < j and the root edge of M k is a border, which proves Lemma 5.4.
We are finally ready to construct the promised involution.
Lemma 5.5. Let O be a set of orientations and let us fix a positive integer n and partitions µ, ν n such that (µ) + (ν) = n − 3. Then, there exists an involution Proof. Let us fix M ∈ M Moreover, the root edge of both M i 1 , and M i 2 is a handle and Combining this with Eq. (20), we obtain that there exists an integer l ∈ {1, 2} such that In other words, there exists ∈ {0, 1} such that M l = τ i τ j M . We define:  Since there are no other cases, we have constructed the required involution, which finishes the proof. Now, we have all necessary ingredients to prove Theorem 5.2.

CONCLUDING REMARKS
We are going to finish this paper by posing some natural questions and remarks related to the combinatorial side of b-conjecture.
6.1. Removing edges in a different order. Note that the function η given by Definition 3.1 is built recursively by the root-deletion procedure, which gives a natural order on the set of edges of a given map M . One can wonder if there are some other, natural ways to define an order on E(M ) which have chances to give an affirmative answer to Conjecture 1.2 using the statistic η as in Definition 3.1, but with respect to the considered order. For instance, the author of this paper together with Féray andŚniady studied [DFS14] a problem of understanding a combinatorial structure of Jack characters, already mentioned in Section 1.4. We proved that a similarly defined "measure of non-orientability", but considered with respect to the uniform random order on E(M ), has many desired properties. Chapuy and the author of this paper constructed in [CD17] a certain directed graph associated with a bipartite quandrangulation q (that is a map with all faces of degree 4), called the Dual Exploration Graph (DEG, for short), which is visiting all faces of q in some particular order. Since maps (not necessarily bipartite) with n edges are in a natural bijection with bipartite quadrangulations with n faces and edges of a given map correspond to faces of an associated quadrangulation, DEG defines also an order on the set of edges of a given map and La Croix suggested [La 15] to use this order to define a measure of non-orientability η with respect to it. We did not study combinatorial properties of the statistic η defined in this way and we leave open the problem wether it gives the correct answer for b-conjecture.
6.2. Unhandled maps and evaluation at b = −1. Corollary 3.3 suggests that unhandled maps are of special interest: indeed, for any measure of non-orientability η and for any partitions µ, ν, τ of a positive integer n, the top-degree coefficient in the polynomial (H η ) τ µ,ν (β) is given by unhandled maps of type (µ, ν; τ ). In particular this top-degree part does not depend on the choice of η, but it does depend on the order of edges we are removing from a given map. Moreover, Eq. (16) ensures that in the case of one-part partition τ = (n), the top-degree coefficient in the polynomial (H η ) τ µ,ν (β) coincides, up to a sign, with the evaluation of this polynomial in β = −1. We can also prove that for any partitions µ, ν of a positive integer n ≥ 2 and for any l ∈ [n − 1] the top-degree coefficient in the polynomial (H η O ) (n−l,l) µ,ν (β) coincides with the evaluation of this polynomial in β = −1 for any set O of orientations (in fact, the main idea of the proof in the case when (µ)+ (ν) ≥ n−3 was given in Lemma 5.3, and the general case is almost the same). Thus, there are natural questions: Question 6.1. Is it true that for any measure of non-orientability η, for any positive integer n, and for any partitions µ, ν, τ n the following equality holds true: (H η ) τ µ,ν (−1) = # M η τ µ,ν;0 ? Question 6.2. Is it true that for any measure of non-orientability η, for any positive integer n, and for any partitions µ, ν, τ n the following equality holds true: (H η ) τ µ,ν (−1) = h τ µ,ν (−1)?
Another interesting direction of the research initiated in this paper is an understanding of the combinatorial structure of an unhandled map of type (µ, ν; τ ) for arbitrary partitions µ, ν, τ of a positive integer n. Note that the set of unhandled maps of a given type is not rooted invariant. However, Proposition 4.1 together with Theorem 4.2 imply that unicellular unhandled maps with the black vertex distribution µ and the white vertex distribution ν are in a bijection with orientable maps with the black vertex distribution µ, the white vertex distribution ν, and the arbitrary face distribution, which clearly are rooted invariant (one can even use a proof of Theorem 4.2 to construct such a bijection recursively). Therefore one can ask the following question: Question 6.3. Is it true that for any given type (µ, ν; τ ) of unhandled maps, there exists some class of maps (orientable?), with the black vertex distribution µ, the white vertex distribution ν, and possibly some additional data (labeling faces?), which is rooted invariant, and which is in some natural bijection with the corresponding set of unhandled maps?

?
We leave all these questions wide open for future research.