Bijection between oriented maps and weighted non-oriented maps

We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.


PROLOGUE
In order to motivate the Reader and to give her some flavor of the results to expect, we shall present now some selected highlights before getting involved in somewhat lengthy definitions. We also deliberately postpone the bibliographic details. 0.1. Maps and orientability. Roughly speaking, a map M = (G, S) is a bicolored graph G which is drawn on a surface S. We require that the connected components of S \ G, called faces, are all homeomorphic to open discs. The set of vertices of G is decomposed into two disjoint sets: the set of white vertices and the set of black vertices. Each edge connects two vertices of opposite colors; multiple edges are allowed. We do not allow isolated vertices. We allow the surface S to be disconnected.
The maps which we consider in the current paper come in the following two flavors: oriented maps (which are drawn on an orientable surface S which comes with some prescribed choice of the orientation, see Figure 1) and non-oriented maps (which are drawn on an arbitrary surface S without FIGURE 1. Example of an oriented map drawn on the torus. The left side of the square should be glued to the right side, as well as bottom to top, as indicated by the arrows. The statement that it is an oriented map means that there is a prescribed consistent choice around each vertex of what clockwise means. any additional structures, see Figure 2). These two flavors are quite distinct, nevertheless some algebraic-combinatorial conjectures which concern Jack polynomials suggest that there is some hypothetical natural one-parameter interpolation between them. To be more specific: it has been conjectured that there exists some measure of non-orientability of a given non-oriented map M which is a hypothetical polynomial weight M (γ) in the deformation parameter γ with the property that if a summation over non-oriented maps is performed, and each map is counted with appropriate multiplicity weight M (γ), the resulting sum becomes a natural interpolation between some generating series of oriented and non-oriented maps.
In the current paper we consider a concrete formula for such a candidate weight, a candidate which will be denoted by mon M (γ), which is an acronym that stands for measure of non-orientability. 0.2. Top-degree of the measure of non-orientability. In order to state the main results of the paper we will not need the definition of the full polynomial mon M (γ) ∈ Q[γ] and we shall restrict ourselves to the leading coefficient of this polynomial which will be denoted by mon top M . We shall present now its definition.
Any non-oriented map M can be equivalently viewed as a ribbon graph, see Figure 3. With this viewpoint each vertex becomes a disc, each edge  becomes a thin ribbon connecting the discs, and each face of the map becomes a connected component of the boundary of the union of the discs and the ribbons.
From a ribbon graph corresponding to M we shall remove all of its edges, one after another, in a uniformly random order. We use the convention that whenever after an edge removal some vertex becomes isolated, we remove this vertex as well.
The following can be viewed as a working definition of the quantity mon top M . Definition 0.1. Assume that M is a non-oriented map. The quantity mon top M is defined to be the probability of the event that in the above process of uniformly random edge removal, at each step the number of faces of the ribbon graph is equal to the number of its connected components.
The link between the above quantity mon top M and the polynomial mon M will be provided later in Proposition 5.4.
Example 0.2. We consider the map M shown in Figure 4; the corresponding ribbon graph is shown in Figure 5.
Consider the case when the ribbon marked {3, 6} is removed first; then the remaining two ribbons form an annulus which has two faces and only one connected component, thus the event considered in Definition 0.1 does not hold.
Consider now the remaining two cases when either the ribbon {1, 5} or the ribbon {2, 4} is removed first; then the remaining two ribbons form the Möbius strip which has one face and one connected component. After yet another ribbon removal the unique remaining ribbon forms a ribbon graph which again consists of one face and one connected component. It follows that the event considered in Definition 0.1 holds.
In this way we found that the probability considered in Definition 0.1 is equal to 2 3 and thus mon top M = 2 3 . 0.3. Rooted maps. The notion of a rooted map takes a different form when we speak about oriented maps than in the case of non-oriented maps.
More specifically, by a rooted oriented map we mean an oriented map in which one edge is decorated. For example, one can take the map from Figure 1 and remove the labels of all edges, except for the edge labeled by the symbol 1, and declare that this edge is decorated.
By a rooted non-oriented map we mean a non-oriented map in which one of the edge-sides is decorated (each edge consists of two edge-sides). For example, one can take the map from Figure 2 and remove the labels of all edge-sides, except for the edge-side labeled by 1, and declare that this edge-side is decorated.   0.4. The first main result. The following is one of the main results of the current paper (for the other one see Theorem 4.3). It states that a summation over oriented maps with arbitrary face structure is equivalent to a weighted summation over non-oriented maps with exactly one face.
Theorem 0.3 (The first main result). For all integers n ≥ 1 the following formal linear combinations of bicolored graphs are equal: where the sum on the left-hand side runs over oriented, unlabeled, rooted, connected maps M 1 with n edges and the sum on the right-hand side runs over non-oriented, unlabeled, rooted maps M 2 with n edges and one face.
It should be stressed that on the left-hand side we impose no restrictions on the face-type of the map M 1 , in particular the number of the faces is arbitrary.
Equation (0.1) is an equality between formal linear combinations of bicolored graphs. Equivalently, it can be viewed as the following statement: for each bicolored graph G, the number of ways in which G can be drawn on (A1) some oriented surface S 1 in such a way that (G, S 1 ) becomes an oriented map is equal to the (weighted) number of ways in which G can be drawn on (A2) some non-oriented surface S 2 in such a way that (G, S 2 ) becomes a non-oriented map with exactly one face.
Our proof of Theorem 0.3 will be bijective. For a fixed bicolored graph G we will find a bijection between: (B1) the set of non-oriented (but orientable) maps (G, S 1 ), together with a choice of an arbitrary linear order on the set of edges of G, and (B2) the set of non-oriented maps (G, S 2 ), together with a choice of a linear order on the set of edges of G with the property that if the edges of a ribbon graph corresponding to the map (G, S 2 ) are removed according to this linear order, then the condition from Definition 0.1 holds true, i.e. at each step the number of faces of the ribbon graph is equal to the number of its connected components.
Roughly speaking, our bijection consists of a number of twists of the ribbons, see Figure 6. The proof of Theorem 0.3 is postponed to Section 3.
One possible motivation for Theorem 0.3 is purely aesthetical. However, there is also another important motivation related to the study of Jack polynomials and Jack characters which will be explored in Section 4 and, in particular, in Sections 4.9 and 4.10. 0.5. Overview. The structure of this paper is twofold. The first part (Sections 1-3) is self-contained and devoted to the proof of the first main result, Theorem 0.3. The second part (Sections 4-5) presents the the motivations, the wider context, the bibliographic details and the applications to Jack characters, in particular to the second main result, Theorem 4.3. FIGURE 6. Twist of a ribbon in a ribbon graph. The details of the notation will be explained in 1.1. Non-oriented maps, informal viewpoint. If we draw an edge of a non-oriented map M = (G, S) with a fat pen (or, alternatively, if we regard an edge of a corresponding ribbon graph) its boundary consists of two edgesides. The maps which we consider in the current paper have labeled edgesides, see Figure 2; in other words each edge carries two labels, one on each of its sides. We also assume that each label is unique.
If we cut the surface S along the edges of the graph G, the map becomes a collection of bicolored polygons, each polygon corresponding to one face of the map, see Figure 7. The original map M can be recovered from this collection of polygons by gluing together pairs of the edges of the polygons; for each pair of glued edges the white (respectively, black) endpoints of the two edges should be glued together.
Thus a non-oriented map can be described alternatively as a collection of bicolored polygons with labeled edges (such as the ones from Figure 7) together with the pair-partition E telling which pairs of edges should be glued together.
1.2.1. Pairings and polygons. A set-partition of a set X is a collection {I 1 , . . . , I r } of pairwise disjoint, non-empty subsets, the union of which is equal to X. A pairing (or, alternatively, pair-partition) of X is a setpartition into pairs.
Let us consider now two pairings B, W of the same set X consisting of 2n elements. We consider the following bicolored, edge-labeled graph L(B, W): • it has n black vertices indexed by the pairs of B and n white vertices indexed by the pairs of W; • its edges are labeled by the elements of X. The extremities of the edge labeled i are the unique pair of B containing i and the unique pair of W containing i. Note that each vertex has degree 2 and each edge has one white and one black extremity. Besides, if we erase the indices of the vertices, it is easy to recover them from the labels of the edges (the index of a vertex is the set of the two labels of the edges incident to this vertex). Thus, in the following we forget the indices of the vertices and view L(B, W) as an edge-labeled graph.
As every vertex has degree 2, the graph L(B, W) can be seen as a collection of polygons. The terminology comes from the fact that it is possible to represent such a triple of pair-partitions as a bicolored graph embedded in a non-oriented (and possibly non-connected) surface. Let us explain how this works.
First, we consider the union of the polygons L(B, W) defined above in Section 1.2.1. The edges of these polygons, that is the elements of the set X, are called edge-sides.
We consider the union of the interiors of these polygons as a (possibly disconnected) surface with a boundary. If we consider two edge-sides, we can glue them: that means that we identify with each other their white extremities, their black extremities, and the edge-sides themselves.
For any pair in the pairing E, we glue the two corresponding edge-sides. In this way we obtain a (possibly disconnected, possibly non-orientable) surface S without boundary. After the gluing, the edges of the polygons form a bicolored graph G embedded in the surface. For instance, with the pairings B and W from Example 1. In general, the graph G has as many connected components as the surface S. Besides, the connected components of S \ G correspond to the interiors of the collection of polygons we are starting from, and, thus, they are homeomorphic to open discs. These connected components are called faces.
This makes the link with the more common definition of maps: usually, a (bicolored) map is defined as a (bicolored) connected graph G embedded in a (non-oriented) surface S in such a way that each connected component of S \ G is homeomorphic to an open disc. It should be stressed that with our definition -contrary to the traditional convention -we do not require the map to be connected.
Note that our maps have labeled edge-sides, and each element of the label set X is used exactly once as a label.
The pairing B (respectively W) indicates which edge-sides share the same corner around a black (respectively white) vertex. This explains the names of these pairings.
This encoding of non-oriented maps by triples of pairings is of course not new. It can for instance be found in [GJ96]; the presentation in that paper is nevertheless a bit different as the authors consider there connected monochromatic maps.
1.3. Face-type. Summation over non-oriented maps. Let 2π 1 , 2π 2 , . . . , 2π ℓ be the numbers of edges of the polygons, or -equivalently -the numbers of edges of the faces of the map. We say that π = (π 1 , . . . , π ℓ ) is the facetype of the collection of polygons or the face-type of the map.
The summation over non-oriented maps M with face-type (n) should be understood as follows: we fix a bicolored polygon L with 2n labeled edges and consider all pair-partitions E of its edges; we sum over the resulting collection of maps M = M(E). We will refer to this kind of summation as conservative summation.
1.4. Edge liberation for non-oriented maps. Sometimes it will be convenient to consider a different way of summing over maps with the face-type (n) in which the arrangement of the labels on the polygon L is not fixed. To be more specific: we consider all maps M = (B ′ , W ′ , E ′ ) where B ′ , W ′ and E ′ are pair-partitions of the same base set X = [2n] with the property that L(B ′ , W ′ ) consists of a single polygon. In order to differentiate this kind of summation, we will call it liberal summation over non-oriented maps with the face-type (n). Proposition 1.3. For each integer n ≥ 1 the following two formal linear combinations of unlabeled maps are equal: where on the left-hand side we consider a liberal summation over nonoriented maps with the face-type (n) and on the right-hand side we consider a conservative summation over non-oriented maps with the face-type (n). On both sides of the equality we remove the labels of the edge-sides of the maps.
Proof. We fix the pair partitions B and W of the same set X = [2n] in such a way that L = L(B, W) is a single polygon. Let us consider the collection where S(2n) denotes the symmetric group which we view as the set of permutations of X. A formal sum of over this collection clearly corresponds to the conservative summation over non-oriented maps M with face type (n), with each summand taken with the multiplicity |S(2n)| = (2n)!; thus this formal sum is equal to 2n times the right-hand side of (1.2).
For a pair partition I of the base set X = [2n] and a permutation π ∈ S(2n) we denote by π(I) the pair-partition of the same base set X defined as follows: for each a, b ∈ S we have To each pair (π, E) we shall associate the map Clearly, the map M ′ has one face and each non-oriented map M ′ with one face on the base set X can be obtained in this way. Furthermore, two pairs (π, E) and (σ, and the permutation σ −1 π leaves each of the partitions B and W invariant; the set of such permutations forms the dihedral group D n of the isometries of a regular polygon with n edges. It follows that there is a bijective correspondence between the preimage of a given map M ′ (we assume that M ′ has a single face) and the dihedral group D n . Therefore a formal sum of M ′ := (B ′ , W ′ , E ′ ) over the collection (1.3) corresponds to the liberal summation with each summand taken with the multiplicity |D n | = 2n; in other words it is equal to the left-hand side of (1.2) multiplied by 2n. This definition is a bit subtle; for example since we do not allow maps having isolated vertices, if some endpoint of E is a leaf, we remove it as well. Removal of an edge might change the topology of the surface on which the map is drawn; for this reason instead of figures of the type presented in Figure 2 it is more convenient to consider for this purpose ribbon graphs, see Figure 3.
For a more rigorous treatment see [DFŚ14, Section 3.6].
1.6. Three kinds of edges. Let a map M with some selected edge E be given (see example in Figure 8). There are three possibilities: • Both sides of the edge E are lying on the boundary of the same face F . This means that if we travel along the boundary of the face F then we visit the edge E twice. Assume that the directions in which we travel twice along the edge E are opposite, see Figure 9a.
In this case the edge E is called straight and we associate to it the weight 1. • Both sides of the edge E are lying on the boundary of the same face F . This means that if we travel along the boundary of the face F then we visit the edge E twice. Assume that the directions in which we travel twice along the edge E are the same, see Figure 9b.
In this case the edge E is called twisted and we associate to it the weight γ. In this case the edge E is called interface and we associate to it the weight 1 2 . The weight given by the above convention will be denoted mon M,E ; we will need this notion much later in Section 5.1. The classification of edges into three types (i.e. straight versus twisted versus interface) will be necessary immediately. For a given map M we will say that a history is an arbitrary linear order ≺ on the set of edges of M. Let E 1 , . . . , E n be the sequence of edges of M, listed according to the linear order ≺. We set M i = M \ {E 1 , . . . , E i }; in other words M 0 , M 1 , . . . , M n is the sequence of maps obtained from M by removing the edges, one by one, in the order prescribed by the history ≺.
Definition 1.5. We say that (M, ≺) is a top-degree pair if M is a nonoriented map and ≺ is a history with the property that each of the maps M 0 , . . . , M n defined above is a top-degree map.
With the above definitions, for a given non-oriented map M of face-type (n), the corresponding quantity mon top M from Definition 0.1 can be reformulated as the probability that for a uniformly random choice of the history ≺, the pair (M, ≺) is a top-degree pair.
We will not make use of the following lemma (but we will make use of its extension, Lemma 5.2). Nevertheless, we decided to state it here because it provides a natural and intuitive interpretation for the notion of top-degree pairs. Lemma 1.6. Let M be a map with n edges and ≺ be a history. We use the notations introduced above.
Then the following conditions are equivalent: • a bridge or a leaf.
Note that an alternative proof of this result (in a wider generality) will be given in Section 5.2.
Proof. Suppose that the condition (B) holds true. It is easy to prove by backward induction that for each 0 ≤ i ≤ n − 1 the map M i is a top-degree map which implies the condition (A).
Assume now that the condition (B) does not hold true. Let i be the maximal number with the property that the edge E i+1 of the map M i is neither a twisted edge, nor a bridge, nor a leaf. Clearly i < n. The reasoning from the previous paragraph shows that the map M i+1 is top-degree. The edge E i+1 is not a bridge, so its endpoints must be located in the same connected component of M i+1 therefore they belong to the same face. Since the edge E i+1 is not a twisted edge of M i , it follows that E i+1 is an interface edge of M i . It follows that the connected component of the map M i which contains the edge E i+1 consists of two faces. In this way we proved that M i is not top-degree and the condition (A) does not hold true.
1.8. Top degree histories and orientable maps. Let M = (B, W, E) be a non-oriented map and let E be one of its edges. We denote by twist E (M) the non-oriented map which is obtained from M by twisting the edge E, see Figure 6. Formally, twist E (M) = (B, W ′ , E) is the map obtained by changing the structure of the pairings of the white vertices, as explained in Figure 6.
Lemma 1.7. Let M be a map, let E be one of its edges and let M 1 := M \E. Assume that M 1 is a top-degree map (respectively, an orientable map).
If E is a bridge or a leaf in the map M, then M is a top-degree map (respectively, an orientable map).
If E is neither a bridge nor a leaf in the map M, then exactly one of the following two maps: M and twist E (M) is a top-degree map (respectively, an orientable map).
The proof is immediate. Proof. The bijection which we will construct will have the following additional property: for each (M, ≺) there exists some subset {e 1 , . . . , e l } of the set of the edges of M such that Φ(M, ≺) = twist e 1 · · · twist e l M, ≺ .
We will use induction with respect to the variable n. For n = 1 both sets (a) and (b) consist of a single element and there is an obvious bijection between them.
Consider the case n ≥ 2. Let (M, ≺) be a top-degree pair. Let E be the first of the edges of M, according to the linear order ≺; let M 1 := M \ E and let ≺ 1 be the restriction of the linear order ≺ to the edges of M 1 ; and let X 1 be the set X with the labels of the edge-sides of E removed.
The pair (M 1 , ≺ 1 ) is also a top-degree pair and thus Φ(M 1 , ≺ 1 ) has been already constructed by the inductive assertion; there exists some set of the edges {e 1 , . . . , e l } of M 1 with the property that M ′ 1 = twist e 1 · · · twist e l M 1 . By twisting the same set of edges in the bigger map M we define M := twist e 1 · · · twist e l M.
There are the following two possibilities.
• The case when the edge E is a bridge or a leaf in the map M.
We define Note that M ′ 1 = M \ E is, by definition, an orientable map and E is a bridge or a leaf in the map M ; thus Lemma 1.7 implies that the map M is orientable, as required.
• The case when the edge E is neither a bridge nor a leaf in the map M.
Since M ′ 1 = M \ E is an orientable map and E is neither a bridge nor a leaf in the map M , Lemma 1.7 implies that exactly one of the maps M and twist E M is orientable; we denote this orientable map by M ′ .
Finally, we define This concludes the inductive construction. It remains to show that the map Φ constructed above has an inverse. The construction of the inverse Φ −1 is quite analogous to that of Φ and we skip it.

ORIENTED MAPS
2.1. Oriented maps. We define an oriented map as a bicolored graph G embedded in an oriented surface S in such a way that each connected component of S \ G is homeomorphic to an open disc. If the number of edges is equal to n, we shall assume that the edges are labeled by the elements of [n] in such a way that each label is used exactly once.
There is a bijective correspondence between such oriented maps with n edges and the set of pairs (σ 1 , σ 2 ), where σ 1 , σ 2 ∈ S(n) are permutations. This correspondence follows from the observation that the structure of such an oriented map is uniquely determined by the counterclockwise cyclic order of the edges around the white vertices (which we declare to be encoded by the disjoint cycle decomposition of the permutation σ 1 ) and by the counterclockwise cyclic order of the edges around the black vertices (which we declare to be encoded by the disjoint cycle decomposition of the permutation σ 2 ). The corresponding oriented map will be denoted by M(σ 1 , σ 2 ).

2.2.
Edge liberation for oriented maps. For σ 1 , σ 2 ∈ S(n) we say that " σ 1 , σ 2 is transitive" if the group generated by σ 1 and σ 2 acts transitively on the underlying set [n]. It is easy to see that the oriented map M(σ 1 , σ 2 ) is connected if and only if σ 1 , σ 2 is transitive.
The following result shows that a summation over oriented, connected maps can be alternatively viewed as a summation over orientable (but not oriented), connected maps.
Proposition 2.2. Let n ≥ 1. Then the following two formal linear combinations of non-oriented unlabeled maps are equal: where the sum on the right-hand side runs over non-oriented maps M = (B, W, E) where B, W, E are arbitrary pair-partitions of the set X = [2n] such that M is orientable and connected.
Let us consider some triple which belongs to (2.2). The permutations σ 1 and σ 2 define an oriented map M(σ 1 , σ 2 ) with the edges labeled by the elements of [n]; in the following we will show how to view this map as a non-oriented map. This will be done by defining the labels associated to all edge-sides of the original map M; these new labels belong to [2n]. In other words, to each edge e of M (i.e., to each label e ∈ [n]) we need to associate an ordered pair of two labels from [2n]; we declare this pair to be f (e, 1), f (e, 2) . More specifically, if we go counterclockwise around the white endpoint of e and read the labels of the edge-sides, then immediately after the label f (e, 1) we should read the label f (e, 2), see Figure 10.
More formally speaking, the above construction corresponds to the non- α FIGURE 10. The convention for defining the labels of the edge-sides. The symbol written directly on the edge is its label; as usual the counterclockwise cycle structure of the edges around the white vertices is given by the permutation σ 1 while the counterclockwise cycle structure of the edges around the black vertices is given by the permutation σ 2 . The two symbols written next to the edge are the labels of the edge-sides.
The resulting non-oriented map M is orientable and connected. Furthermore, each such an orientable, connected, non-oriented map can be obtained in this way in 2 · n! ways (the factor 2 comes from the two possible ways of choosing the orientation of the map, the factor n! counts the possible choices of the labeling of the edges of the oriented map from which we start).
The comparison of the multiplicities arising from the two ways of interpreting the summation over (2.2) concludes the proof.

PROOF OF THE FIRST MAIN RESULT, THEOREM 0.3
Proof of Theorem 0.3. Recall that the left-hand side of (0.1) runs over oriented, unlabeled, rooted, connected maps M 1 with n edges. Each such a map can become a labeled map in (n − 1)! distinct ways as follows: we attach to the root edge the label 1 and we attach the remaining labels from [n] \ {1} to the other edges. Thus the left-hand side of (0.1) is equal to which is viewed as linear combination of unlabeled maps.
Edge liberation for oriented maps (Proposition 2.2) implies that (3.1) is equal to where the sum runs over non-oriented maps M over the base set X = [2n] such that M is orientable and connected. Our analysis of the left-hand side of (0.1) is now completed and we shall turn to its right-hand side.
Recall that the sum on the right-hand side of (0.1) runs over non-oriented, unlabeled, rooted maps M 2 with n edges and one face. Note that there is a canonical way of defining the labels on the edge-sides of this map, as follows. We perform a walk along the boundary of the unique face of map, with the first step along the decorated edge-side in the direction from its black to its white extremity. We attach the label 1 to this decorated edgeside, and we continue to label the edge-sides in the order in which we visit them with the remaining elements of the label set [2n]\{1}. This means that the sum on the right-hand side of (0.1) is in fact a conservative summation over non-oriented maps with face-type (n).
From Definition 0.1 it follows now that the right-hand side of (0.1) is equal to where the first sum is a conservative sum over non-oriented maps with facetype (n) and the second sum runs over the histories such that (M, ≺) is a top-degree pair. We apply edge liberation for non-oriented maps (Proposition 1.3); in this way (3.3) takes the form where the first sum is this time a liberal sum over non-oriented maps with face-type (n).
We apply the bijection provided by Theorem 1.8; in this way we see that (3.4) is equal to where the sum runs over all connected, orientable non-oriented maps over the base set [2n].
Since (3.2) and (3.5) are equal as formal linear combinations of unlabeled graphs, this concludes the proof.

BACKGROUND AND HISTORY OF THE RESULT: JACK CHARACTERS
In this section we present the background and the motivations for Theorem 0.3, in particular the problems related to Jack characters.

Jack characters.
4.1.1. Jack polynomials. Henry Jack [Jac71] introduced a family J (α) π (indexed by an integer partition π) of symmetric functions which depend on an additional parameter α. During the last forty years, many connections of these Jack polynomials with various fields of mathematics and physics were established: it turned out that they play a crucial role in understanding Ewens random permutations model [DH92], generalized β-ensembles and some statistical mechanics models [OO97], Selberg-type integrals [Kan93], certain random partition models [BO05], and some problems of algebraic geometry [Oko03], among many others. Better understanding of Jack polynomials is also very desirable in the context of generalized β-ensembles and their discrete counterpart model [DF16b]. Jack polynomials are a special case of the celebrated Macdonald polynomials which "have found applications in special function theory, representation theory, algebraic geometry, group theory, statistics and quantum mechanics" [GR05]. 4.1.2. Dual combinatorics of Jack polynomials. Lassalle [Las08,Las09a] initiated investigation of a kind of dual combinatorics of Jack polynomials. More specifically, one expands Jack polynomial in the basis of power-sum symmetric functions: The above sum runs over partitions π such that |π| = |λ|. The coefficient θ (α) π (λ) is called unnormalized Jack character; with the right choice of the normalization it becomes the normalized Jack character Ch π (λ) (the details of this relationship will be given in Definition 4.1). An interesting feature of Jack characters is that for the special choice of the deformation parameter α = 1 they coincide with the usual characters of the symmetric groups.
The above approach is referred to as dual because one fixes π and views the character λ → Ch π (λ) as a function of the Young diagram λ, opposite to the usual approach in the representation theory where one usually fixes the irreducible representation (the diagram λ) and views the character as a function of the conjugacy class (the partition π). In the context of the representation theory of the symmetric groups (which corresponds to the special case α = 1) this dual approach was initiated by Kerov and Olshanski [KO94] and it soon turned out to be highly successful; its applications include, for example, random Young diagrams [IO02,Bia98,Śni06]. Lassalle [Las08,Las09a] adapted this idea to the framework of Jack characters.
Jack characters Ch π (λ) are in the focus of the current paper. Our motivations for investigating them are threefold. Firstly, since Jack characters are related to Jack polynomials, a better understanding of the former might shed some light on the latter. Secondly, they can be used in order to investigate some natural deformations of classical random Young diagrams [Śni16,DŚ16]. Thirdly, numerical data [Las09b] as well as some partial theoretical results [Las08, Las09a, DFŚ14,Śni15,Śni16] indicate that they might have a rich algebraic-combinatorial or representation-theoretic structure.
Our ultimate goal would be to find some convenient closed formula for Jack characters. This goal is beyond our reach; a more modest goal would be to find a closed formula for the dominant part of Jack characters in the suitable asymptotic scaling. We shall address this issue later on. 4.1.3. Definition of Jack characters. In order for this dual approach to be successful (both with respect to the usual characters of the symmetric groups and for the Jack characters) one has to choose the most convenient normalization constants. In the current paper we will use the normalization introduced by Dołęga and Féray [DF16b] which offers some advantages over the original normalization of Lassalle. Thus, with the right choice of the multiplicative constant, the unnormalized Jack character θ (α) λ (π) from (4.1) becomes the normalized Jack character Ch (α) π (λ). Regretfully, their definition is quite technical and not very enlightening. On the bright side, this definition is not relevant for the purposes of the current paper and the Readers faint at heart are cordially invited to fast forward to Section 4.1.4.
Definition 4.1. Let α > 0 be given and let π be a partition. The normalized Jack character Ch π (λ) is given by: is the standard numerical factor. Jack character Ch π (λ) depends on the deformation parameter α, but to keep the notation light we shall usually make this dependence implicit. 4.1.4. The deformation parameter. In order to avoid dealing with the square root of the variable α, we introduce an indeterminate A such that In this way Jack character Ch π (λ) ∈ Q [A, A −1 ] becomes a Laurent polynomial in the indeterminate A. This is the viewpoint which we will usually take in this paper, with the exception of Sections 4.3.1 and 4.3.2 where A will be some fixed real number.

Bicolored graphs and their embeddings.
An embedding F of a bicolored graph G to a Young diagram λ is a function which maps the set V • (G) of white vertices of G to the set of columns of λ, which maps the set V • (G) of black vertices of G to the set of rows of λ, and maps the edges of G to boxes of λ, see Figure 11. We also require that an embedding preserves the relation of incidence, i.e., a vertex V and an incident edge E should be mapped to a row or column F (V ) which contains the box F (E). We denote by N G (λ) the number of such embeddings of G to λ. Quantities N G (λ) were introduced by Féray and the second-named author [FŚ11a] and they proved to be very useful for studying various asymptotic and enumerative problems of the representation theory of symmetric groups [FŚ11a,FŚ11b,DFŚ10].
Definition 4.2. For a bicolored graph G and a Young diagram λ we define the normalized number of embeddings [Śni15] which is a Laurent polynomial in A: A very similar quantity denoted by N  √ 2 , ±1, ± √ 2 which correspond to α ∈ 1 2 , 1, 2 , Jack characters admit closed formulas in terms of embeddings of certain bicolored maps. Formulas of this type are called Stanley formulas after Richard Stanley, who found such a formula for α = 1 as a conjecture [Sta06]. We shall review them in the following. 4.3.1. Stanley formula for α = 1 and oriented maps. In the special case of A = ±1 which corresponds to α = 1, Jack polynomials coincide (up to simple multiplicative constants) with Schur polynomials. Using this fact one can show that in this special case the Jack character Ch A:=1 π coincides with the (suitably normalized) character of the symmetric group; for the details see the work of Lassalle [Las09a] (who used a different normalization) as well as the work of Dołęga and Féray [DF16b]. For this reason, for A = 1 Jack characters have a much richer algebraic and representation-theoretic structure than for a generic value of A.
In particular, it has been observed in [FŚ11a] that a certain formula conjectured by Stanley [Sta06] and proved by Féray [Fér10] for the normalized characters of the symmetric groups can be expressed as the sum over all oriented bicolored maps M with face-type π, where ℓ(π) denotes the number of parts of the partition π.

4.3.2.
Stanley formula for α ∈ {2, 1 2 } and non-oriented maps. In the special case when A = ± √ 2 and α = 2 (respectively, A = ± 1 √ 2 and α = 1 2 ) Jack polynomials coincide with zonal polynomials (respectively, symplectic zonal polynomials). Thanks to this additional structure it has been proved in a joint work of the second-named author with Féray [FŚ11b] that where the sums run over all non-oriented maps M with face-type π, as in Section 1.3. The reader should be advised that the notations and the normalizations in [FŚ11b] are a bit different; the link between the statements above and the results of [FŚ11b] is given in [DFŚ14, Section 5].
4.4. How to prove a closed formula for Jack characters? Orientability generating series. Jack characters admit an abstract characterization [Śni15, Theorem 1.7] which was found in a recent paper of the secondnamed author (see also Theorem A.2 of Féray in an appendix to the same paper [Śni15]). This abstract characterization opens the following path toward proving a closed formula for Jack characters: in the first step one should guess the right formula, and in the second step one should verify that it indeed fulfills the aforementioned defining properties of Jack characters.
How to make the first step and to guess a closed formula for Jack characters? The three Stanley formulas (4.4), (4.5), (4.6) might suggest that the hypothetical formula for Jack characters in the generic case should be of the form (4.7) Ch π (λ) = (−1) ℓ(π) where the sum should run over non-oriented maps with face-type π. In the above formula weight M ∈ Q [A, A −1 ] is some hypothetical quantity which measures the non-orientability of the map M [DFŚ14, Conjecture 1.1].
A joint work of the second-named author with Dołęga and Féray [DFŚ14] presents an attempt to guess the exact form of this hypothetical quantity weight M , an attempt which was based on a reverse-engineering of the results of Lassalle [Las08]. Our candidate quantity, denoted mon M (which is an acronym for measure of non-orientability) was defined as follows: we remove the edges from the ribbon graph of M in a uniformly random order; to each edge which is about to be removed we associate a factor which is related to the topological way in which this edge is attached to the remaining edges (i.e., the edges which have not been removed yet) as we discussed in Section 1.6. The quantity mon M is defined as the expected value of the product of the aforementioned factors. The details of this construction will be recalled in Section 5.3. This weight mon M gives rise to the orientability generating series [DFŚ14, Section 1.10] which is defined in analogy to (4.7) as where the sum runs over non-oriented maps with face-type π.
4.5. Stanley polynomials. In order to be able to speak about the degree of some functions on the set of Young diagrams we will need the notion of Stanley polynomials. The content of this section is an abridged and less formal version of [Śni15, Section 1.10].
This concept is illustrated in Figure 12.
4.5.2. The deformation parameter γ. We usually view Jack polynomials as functions of the parameter α and Jack characters as functions of another parameter A = √ α. However, it is convenient to consider yet another deformation parameter As we shall see, several quantities can be expressed as polynomials in γ. be a polynomial in 2ℓ + 1 variables. Suppose that the equality -with the substitution (4.9) for the variable γ -holds true for all choices of P , Q and A = 0 for which the multirectangular diagram P ′ × Q ′ is well-defined. Then we say that St ℓ is the Stanley polynomial for F . The above definition is not very precise; on the formal level one should consider Stanley polynomial as an element of some inverse limit for ℓ → ∞; for the details we refer to [Śni15, Section 1.10].
Stanley polynomials are a perfect tool for studying asymptotic questions in the setup when the Young diagram P ′ × Q ′ tends to infinity. In particular, we say that F : Y → Q [A, A −1 ] is a function of degree (at most) d if the corresponding Stanley polynomial is of degree (at most) d.

Stanley polynomials for
Ch π and Ch π . It is a highly non-trivial result of Dołęga and Féray [DF16b, Corollary 3.5] (see also [Śni15, Theorem 2.15, Corollary 2.11] for details how to adapt their result to our setup) that a Stanley polynomial exists for each Jack character Ch π and that is of degree at most |π| + ℓ(π). For example, One can show that the Stanley polynomial for the orientability generating series Ch π exists and it is also of degree |π| + ℓ(π); the proof is postponed to Lemma 5.5. 4.6. Orientability generating series versus Jack character. The initial prediction of the authors of [DFŚ14] was that the quantity mon M is the right guess for the hypothetical weight M and thus Ch π = Ch π . Regretfully, this turns out to be not the case [DFŚ14, Section 7] and Ch π = Ch π in general.
This might seem as the end of the story and an example of a failed research, nevertheless the orientability series Ch π appeared to predict the properties of the Jack character Ch π suspiciously well. The latter statement was supported heuristically in the following two ways.
• Firstly, computer-assisted comparison of the coefficients of Stanley polynomials for the character Ch π with their counterpart for the orientability generating series Ch π indicates that a lot of them coincide for several concrete examples. • Secondly, in the case when π = (n) is a partition with only one part, a comparison of (i) the contribution to the orientability generating series Ch n which comes from the maps M with small genus g ≤ 3 2 with (ii) its counterpart for Jack character Ch n indicate a full match [CJ16,Section 5]. For (i) a closed formula found by the first named author [CJ16] is available. The comparison can be either performed numerically for small values of n using the data provided by Lassalle [Las09b], or by comparing the aforementioned closed formula for (i) with Lassalle's conjectural closed formula [Las09a, Section 11]; they turn out to be identical. 4.7. The second main result: the top-degree of Jack character. One of the main results of the current paper is the following positive result. Above, Ch top n denotes the top-degree of Jack character Ch n . More explicitly, Stanley polynomial for Ch n is known to be of degree n + 1; the homogeneous part of this Stanley polynomial of degree n + 1 defines a function on the set of Young diagrams which will be denoted Ch top n . Analogously, the top-degree Ch top n of the orientability generating series is defined as the homogeneous part of Ch n of degree n + 1.
In the remaining part of this section we will explain the relationship between the two main results of the current paper: Theorem 0.3 and Theorem 4.3. 4.8. Two formulas for the top-degree part of Jack characters. The following result was proved by the second-named author using a modified version of the strategy which we outlined in Section 4.4: guess the right closed formula and then verify that it satisfies some abstract characterization (this abstract characterization of Ch top n turns out to be much more complex than the analogous characterization of Ch n ). where the sum runs over oriented, unlabeled, rooted, connected maps with n edges and with arbitrary face-type.
Our proof of Theorem 4.3 will be based on the following simple observation: our main bijective result (Theorem 0.3) shows equality between the right-hand sides of (4.8) and (4.11). The missing details of the proof will be provided in Section 5.6. 4.9. Top-degree of Jack characters: the real story. Motivations for Theorem 0.3. The (yet unpublished at the time) bijective result of the current paper (Theorem 0.3) preceded the proof of the above-mentioned closed formula for the top-degree of Jack characters from Theorem 4.4. Indeed, the key difficulty in Theorem 4.4 was to guess the right formula and this difficulty was overcome by converting a conjectural formula based on the ideas from [DFŚ14] to a more convenient form thanks to Theorem 0.3. For more discussion on this topic see below.
4.10. Outlook into the future. The Reader interested in Jack characters may wonder: why bother now proving the second main result of the current paper, Theorem 4.3. After all, a simpler closed formula for Ch top n (Theorem 4.4) was already available; the formula (4.10) has served its duty as a source of heuristics and can be retired now.
However, our ultimate goal, finding a closed formula for Jack characters Ch n (and, more generally, Ch π ), has not been completed. The latter is currently far beyond our reach; a more modest partial accomplishment would be to find a closed formula for the next term in the asymptotic expansion of Jack character Ch n after its top-degree part Ch top n . Once the right candidate formula is found by some heuristic means, the machinery from [Śni15] could be probably relatively easily adapted in order to prove that such a candidate formula indeed holds true.
Regretfully, there is a missing key element to this approach: we have no good candidate formula for such a sub-dominant part of Ch n of degree n − 1 (see [Śni15,Section 1.17]). Probably the simplest approach would be to deduce or extrapolate such a formula based on existing formulas for the asymptotically dominant part Ch top n . Unfortunately, the final formula (4.11) for Ch top n does not seem to offer any hints how to extrapolate it into the sub-dominant regime.
On the other hand, our original starting point, the orientability generating series Ch n given by (4.8), offers immediately a candidate formula for such a sub-dominant part. We have to admit: it is a candidate formula which gives slightly wrong predictions [DFŚ14, Section 7], but it is nevertheless a good starting point for some better formula.
Suppose that this happens to be indeed the case and someone, someday finds such a better formula based on the concept of the orientability generating series. In such a scenario some modified version of the bijection behind our main bijective result (Theorem 0.3) might become handy in order to convert this hypothetical formula into a more convenient form, just like we used this bijection in the current paper in order to transform the right-hand side of (4.8) into more convenient the right-hand side of (4.11).
For more discussion on this topic see [Śni15, Section 1.17].
4.11. b-conjecture. In this section all maps are connected, and rooted, that is they posses a marked, oriented corner incident to some black vertex (that is, an angular region around black vertex v delimited by two consecutive edges attached to v).
Goulden and Jackson [GJ97] introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; 1, 1 + β) with an additional parameter β that might be interpreted as a continuous deformation of the rooted bicolored maps generating series. Indeed, it has the property that for β ∈ {0, 1} it specializes to rooted, orientable (for β = 0) or general, i.e. orientable or not (for β = 1) bicolored maps generating series. Goulden and Jackson made the following conjecture: coefficients of ψ (indexed by three partitions of the same size) are polynomials in β with positive integer coefficients that can be written as a multivariate generating series of rooted, general bicolored maps, where the exponent of β is an integer-valued statistics that in some sense "measures the non-orientability" of the corresponding bicolored map.
This b-conjecture has not yet been proved, but some progress towards determining both algebraic and combinatorial properties of the coefficients in question has been made, and the work is ongoing. Dołęga and Féray [DF16a] proved recently that all coefficients of ψ are polynomials in β with rational coefficients. Based on this result, Dołęga [Doł16] found a combinatorial interpretation of the top-degree of coefficients of ψ indexed by two arbitrary partitions µ, ν ⊢ n, and one partition consisting of only one part (n), which conjecturally should be given by certain maps with only one face. An interesting phenomenon is that, similarly as in our Theorem 0.3, the top-degree part found by Dołęga is given by orientable maps with the black, and white, respectively, vertex degrees given by partitions µ, and ν, respectively, and arbitrary face structure, and at the same time it is given by certain maps (called unhandled) with a unique face, and black and white vertex degrees given by partitions µ, and ν, respectively.
We cannot resist to state that there must be a deep connection between these problems, and understanding it would be of great interest.

TOP-DEGREE OF THE ORIENTABILITY GENERATING SERIES
In this section we will present the details of the definition of the polynomial mon M (γ), its relationship to the quantity mon top M introduced in Definition 0.1, and the missing details of the proof of Theorem 4.3.
5.1. Weight associated to a map with a history. We continue the discussion from Section 1.7.
Let E 1 , . . . , E n be the sequence of edges of a non-oriented M, listed according to the linear order ≺. We set M i = M \ {E 1 , . . . , E i } and define This quantity (5.1) can be interpreted as follows: from the map M we remove (one by one) all the edges, in the order specified by the history. For each edge which is about to be removed we consider its weight mon M i ,E i+1 relative to the current map (recall that the factor mon M i ,E i+1 was defined in Section 1.6 and it depends on the type of the edge E i+1 in the map M, i.e. straight versus twisted versus interface).

5.2.
The top-degree of mon M,≺ . The following result provides some crude information about the polynomial mon M,≺ (γ). Here we use the term genus with a small abuse of notation; usually it is used only for orientable surfaces while we use it also for a non-orientable connected map by setting The remaining part of this section is devoted to the investigation of the corresponding leading coefficient The following result is an extension of Lemma 1.6; its only new component is condition (C). If the above conditions hold true then the leading coefficient of the polynomial mon M,≺ is given by This quantity can be interpreted as the mean value of the weight associated to the map M equipped with a randomly selected history (with all histories having equal probability). We call mon M the measure of non-orientability of the map M. is removed first the corresponding weight is equal to mon M,≺ = 1 · 1 2 · 1. For the remaining 4 histories the corresponding weight is equal to mon M,≺ = γ · γ · 1. Finally, mon M = 2 × 1 · 1 2 · 1 + 4 × γ · γ · 1 6 . Proof. This is an immediate application of Lemma 5.2.
5.4. Orientability generating series. We recall that the orientability generating series was defined in (4.8) as a weighted sum of the normalized numbers of embeddings, with the weight given by the polynomial mon M described above: (5.6) Ch π (λ) := (−1) ℓ(π) where the sum is a conservative summation over non-oriented maps with the face-type π.
Proof. It is relatively easy to show that for any bicolored graph G the corresponding normalized number of embeddings λ → N G (λ) has a Stanley polynomial which is homogeneous of degree |V (G)|. Proposition 5.4 provides an upper bound on the degree of the polynomial mon M .
By combining the above two observations it follows that the Stanley polynomial for the product mon M N M exists and its degree is bounded from above by |E(M)| + |F (M)| = |π| + ℓ(π), as required.
5.5. The top-degree of the orientability generating series.
Corollary 5.6. For any integer n ≥ 1 the top-degree homogeneous part of the orientability generating series is given by where the sum on the right-hand side runs over non-oriented maps with face-type (n).
Proof. By Lemma 5.5 Stanley polynomial for Ch n is of degree (at most) n+ 1; out goal now is to extract its homogeneous part of degree n + 1. This can be done by revisiting the proof of Lemma 5.5 and using Proposition 5.4. 5.6. Proof of Theorem 4.3.
Proof of Theorem 4.3. The left-hand side of (4.10) is given by Theorem 4.4. The right-hand side of (4.10) is given by Corollary 5.6. Now it is enough to apply Theorem 0.3 to show that they are equal.