A simple model of trees for unicellular maps

A simple model of trees for unicellular maps

Guillaume Chapuy, Valentin Féray and Éric Fusy CNRS, LIAFA - UMR 7089, Université Paris 7, 75205 Paris Cedex, France
guillaume.chapuy@liafa.univ-paris-diderot.fr
CNRS, LaBRI - UMR 5800, Université de Bordeaux, 33405 Talence Cedex, France
feray@labri.fr
CNRS, LIX - UMR 7161, École Polytechnique, 91128 Palaiseau Cedex, France
fusy@lix.polytechnique.fr
Abstract.

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the “recursive part” of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure.

All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and the Goupil-Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-3-constellations.

Key words and phrases:
one-face map, Stanley character polynomial, bijection, Harer-Zagier formula, Rémy’s bijection.
First and third author partially supported by the ERC grant StG 208471 – ExploreMaps.
Second author partially support by ANR project PSYCO

1. Introduction

A unicellular map is a connected graph embedded in a surface in such a way that the complement of the graph is a topological disk. These objects have appeared frequently in combinatorics in the last forty years, in relation with the general theory of map enumeration, but also with the representation theory of the symmetric group, the study of permutation factorizations, the computation of matrix integrals or the study of moduli spaces of curves. All these connections have turned the enumeration of unicellular maps into an important research field (for the many connections with other areas, see [22] and references therein; for an overview of the results see the introductions of the papers [12, 2]) The counting formulas for unicellular maps that appear in the literature can be roughly separated into two types.

The first type deals with colored maps (maps endowed with a mapping from its vertex set to a set of colors). This implies “summation” enumeration formulas (see [19, 29, 24] or paragraph 3.4 below). These formulas are often elegant, and different combinatorial proofs for them have been given in the past few years [23, 17, 29, 24, 2]. The issue is that some important topological information, such as the genus of the surface, is not apparent in these constructions.

Formulas of the second type keep track explicitly of the genus of the surface; they are either inductive relations, like the Harer-Zagier recurrence formula [19], or are explicit (but quite involved) closed forms, like the Lehman-Walsh [32] and the Goupil-Schaeffer [18] formulas. From a combinatorial point of view, these formulas are harder to understand. A step in this direction was done by the first author in [12] (this construction is explained in subsection 2.2), which led to new induction relations and to new formulas. However the link with other formulas of the second type remained mysterious, and [12] left open the problem of finding combinatorial proofs of these formulas.

The goal of this paper is to present a new bijection between unicellular maps and surprisingly simple objects which we call C-decorated trees (these are merely plane trees equipped with a certain kind of permutation on their vertices). This bijection, presented in Section 2, is based on the previous work of the first author [12]: we explicitly describe the “recursive part” appearing in this work. As a consequence, not only can we reprove all the aforementioned formulas in a bijective way, thus giving the first bijective proof for several of them, but we do that in a unified way. Indeed, C-decorated trees are so simple combinatorial objects that all formulas follow from our bijection as an immediate corollary or easy exercise, as we will see in Section 3.

Another interesting application of this bijection, studied in Section 4, is a new explicit way of computing the so-called Stanley character polynomials. The latter are nothing but the evaluation of irreducible characters of the symmetric groups, properly normalized and parametrized. Indeed, in a previous work [14], the second author expressed these polynomials as a generating function of (properly weighted) unicellular maps. Although we do not obtain a “closed form” expression (there is no reason to believe that such a form exists!), we express Stanley character polynomials as the result of a term-substitution in free cumulants, which are another meaningful quantity in representation theory of symmetric groups.

In Section 5 we discuss the possibility of applying our tools to -constellations. This notion is a generalization of the notion bipartite maps introduced in connection with the study of factorizations in the symmetric group. A remarkable formula by Poulalhon and Schaeffer [26] (proved with the help of algebraic tools) suggests the possibility of a combinatorial proof using technique similar to ours. Although our bijection does not apply to these objects, we present two partial results in this direction, in the case of -constellations. One of them is an enumeration formula for a related family of objects that we call quasi--constellations, that turns out to be surprisingly similar to the Poulalhon-Schaeffer formula.

2. The main bijection

2.1. Unicellular maps and C-decorated trees

We first briefly review some standard terminology for maps.

A map of genus is a connected graph embedded on a closed compact oriented111Maps can also be defined on non-orientable surfaces. However, for non-orientable surfaces, only an asymptotic version [4] of the bijection of [12] has been discovered so far. Since the (complete) bijection of [12] is an essential building block of all the results presented in the present paper, we will not talk of non-orientable surfaces in this paper. surface of genus , such that is a collection of topological disks, which are called the faces of . Loops and multiple edges are allowed. The (multi)graph is called the underlying graph of and its underlying surface. Two maps that differ only by an oriented homeomorphism between the underlying surfaces are considered the same. A corner of is the angular sector between two consecutive edges around a vertex. A rooted map is a map with a marked corner, called the root; the vertex incident to the root is called the root-vertex. By convention, the map with one vertex and no edge (of genus ) is considered as rooted at its unique vertex (the entire sector around the vertex is considered as a corner, which is the root).From now on, all maps are assumed to be rooted (note that the underlying graph of a rooted map is naturally vertex-rooted). A unicellular map is a map with a unique face. The classical Euler relation ensures that a unicellular map with edges has vertices. A plane tree is a unicellular map of genus .

A rotation system on a connected graph consists in a cyclic ordering of the half-edges of around each vertex. Given a map , its underlying graph is naturally equipped with a rotation system given by the clockwise ordering of half-edges on the surface in a vicinity of each vertex. It is well-known that this correspondence is -to-, i.e., a map can be considered as a connected graph equipped with a rotation system (thus, as a purely combinatorial object). We will take this viewpoint from now on.

We now introduce a new object called C-decorated tree.

A cycle-signed permutation is a permutation where each cycle carries a sign, either or . A C-permutation is a cycle-signed permutation where all cycles have odd length, see Figure 1(a). For each C-permutation on elements, the rank of is defined as , where is the number of cycles of . Note that is even since all cycles have odd length. The genus of is defined as . A C-decorated tree on edges is a pair where is a plane tree with edges and is a C-permutation of elements. The genus of is defined to be the genus of . Note that the vertices of can be canonically numbered from to (e.g., following a left-to-right depth-first traversal), hence can be seen as a permutation of the vertices of , see Figure 1(c). The underlying graph of is the (vertex-rooted) graph with edges that is obtained from by merging into a single vertex the vertices in each cycle of (so that the vertices of correspond to the cycles of ), see Figure 1(d).

Definition 1.

For nonnegative integers, denote by the set of unicellular maps of genus with edges; and denote by the set of C-decorated trees of genus with edges.

For two finite sets and , we denote by their disjoint union and by the set made of disjoint copies of . Besides, we write if there is a bijection between and . Our main result will be to show that , with a bijection which preserves the underlying graphs of the objects.

2.2. Recursive decomposition of unicellular maps

In this section, we briefly recall a combinatorial method developed in [12] to decompose unicellular maps.

Proposition 1 (Chapuy [12]).

For , denote by the set of maps from in which a set of vertices is distinguished. Then for and ,

 (1) 2g Eg(n)≃E(3)g−1(n)+E(5)g−2(n)+E(7)g−3(n)+⋯+E(2g+1)0(n).

In addition, if and are in correspondence, then the underlying graph of is obtained from the underlying graph of by merging the vertices in into a single vertex.

We now sketch briefly the construction of [12]. Although this is not really needed for the sequel, we believe that it gives a good insight into the objects we are dealing with (readers in a hurry may take Proposition 1 for granted and jump directly to subsection 2.3). We refer to [12] for proofs and details.

We first explain where the factor comes from in (1). Let be a rooted unicellular map of genus with edges. Then has corners, and we label them from to incrementally, starting from the root, and going clockwise around the (unique) face of (Figure 2). Let be a vertex of , let be its degree, and let be the sequence of the labels of corners incident to it, read in clockwise direction around starting from the minimal label . If for some lying in (i.e. in the set of integers between and , including and ), we have , we say that the corner of labelled by is a trisection of . Figure 2(a) shows a map of genus two having four trisections. More generally we have:

Lemma 2 ([12]).

A unicellular map of genus contains exactly trisections. In other words, the set of unicellular maps of genus with edges and a marked trisection is isomorphic to .

Now, let be a trisection of of label , and let be the vertex it belongs to. We denote by the corner of with minimum label and by the corner with minimum label among those which appear between and clockwise around and whose label is greater than . By definition of a trisection, is well defined. We then construct a new map , by slicing the vertex into three new vertices using the three corners as in Figure 2(b). We say that the map is obtained from by slicing the trisection . As shown in [12], the new map is a unicellular map of genus . We can thus relabel the corners of from to , according to the procedure we already used for . Among these corners, three of them, say are naturally inherited from the slicing of , as on Figure 2(b). Let be the vertices they belong to, respectively. Then the following is true [12]: In the map , the corner has the smallest label around the vertex , for . For , either the same is true, or is a trisection of the map .

We now finally describe the bijection promised in Proposition 1. It is defined recursively on the genus, as follows. Given a map with a marked trisection , let be obtained from by slicing , and let be defined as above for . If has the minimum label at , set , which is an element of . Otherwise, let , and set . Note that this recursive algorithm necessarily stops, since the genus of the map decreases and since there are no trisections in unicellular maps of genus (plane trees). Thus this procedure yields recursively a mapping that associates to a unicellular map with a marked trisection another unicellular map of a smaller genus, with a set of marked vertices (namely the set of vertices which have been involved in a slicing at some point of the procedure). The set of marked vertices necessarily has odd cardinality, as easily seen by induction. Moreover, it is clear that the underlying graph of coincides with the underlying graph of in which the vertices of have been identified together into a single vertex. One can show that is a bijection [12], with an explicit inverse mapping.

2.3. Recursive decomposition of C-decorated trees

We now propose a recursive method to decompose C-decorated trees, which can be seen as parallel to the decomposition of unicellular maps given in the previous section. Denote by (resp. ) the set of C-permutations on elements (resp. on elements and of genus ). A signed sequence of integers is a pair where is an integer sequence and is a sign, either or . We will often write signed sequences with the sign preceding the sequence as a exponent, such as .

Lemma 3.

Let be a finite non-empty set of positive integers. Then there is a bijection between signed sequences of distinct integers from —all elements of being present in the sequence— and C-permutations on the set . In addition the C-permutation has one cycle if and only if the signed sequence has odd length and starts with its minimal element.

Proof.

The bijection is illustrated in Figure 3. Starting from a signed sequence , decompose into blocks according to the left-to-right minimum records. Then treat the blocks successively from right to left. At each step, if the treated block has odd length, turn into the signed cycle ; if has even length, move the second element of out of , insert it at the end of the block preceeding , and then turn into the signed cycle . Update the block-decomposition (according to left-to-right minimum records) on the left of (it is very simple, two cases occur: if is the minimum of the elements on the left of , it occupies a single block; if not, is integrated at the end of the block on the left of ). At the end of the right-to-left traversal, the last treated block has odd length, we produce as the last signed cycle. The output is the -permutation made of all the signed cycles that have been produced during the traversal.

Conversely (read Figure 3 from right to left and bottom to top), starting from a -permutation , write as the ordered list of its signed cycles, each cycle starting with its minimal element, and the cycles being ordered from left to right such that the minimal elements are in descending order. Record the sign of the leftmost signed cycle , and turn into the block . Then treat the signed cycles from left to right (starting with the second one). At each step, let be the treated signed cycle and let be the block to the left of . Turn into the block , and in case , move the last element of to the second position of (this possibly makes empty, in which case we erase of the current list of blocks). At the end, we get an ordered list of blocks, which can be seen as a sequence . The output is the signed sequence .

It is easy to see that the two mappings (from signed sequences to -permutations) and (from -permutations to signed sequences) are inverse of each other; indeed these two mappings consist of a sequence of steps that operate on hybrid structures (a sequence of blocs followed by a sequence of signed cycles, these all start with their minimal element, and the minimal elements decrease from left to right), each step of (resp. ) increases (resp. decreasing) by the number of signed cycles in the hybrid structure, and the step of with signed cycles is the inverse of the step of with signed cycles. ∎

An element of a C-permutation is called non-minimal if it is not the minimum in its cycle. Non-minimal elements play the same role for C-permutations (and C-decorated trees) as trisections for unicellular maps. Indeed, a C-permutation of genus has non-minimal elements (compare with Lemma 2), and moreover we have the following analogue of Proposition 1:

Proposition 4.

For , denote by the set of C-decorated trees from in which a set of cycles is distinguished. Then for and ,

 2g Tg(n)≃T(3)g−1+T(5)g−2+T(7)g−3+⋯+T(2g+1)0.

In addition, if and are in correspondence, then the underlying graph of is obtained from the underlying graph of by merging the vertices corresponding to cycles from into a single vertex.

Proof.

For let be the set of C-permutations from where a subset of cycles are marked. Let be the set of C-permutations from where a non-minimal element is marked. Note that since a C-permutation in has non-minimal elements.

We now claim that . Indeed starting from , write the signed cycle containing the marked element of as a signed sequence beginning with and apply Lemma 3 to this signed sequence: this produces a collection of signed cycles of odd length, which we take as the marked cycles.

We have thus shown that . Since by definition , we conclude that . The statement on the underlying graph just follows from the fact that the procedure in Lemma 3 merges the marked cycles into a unique cycle. ∎

2.4. The main result

Theorem 5.

For each non-negative integers and we have

 2n+1Eg(n)≃Tg(n).

In addition the cycles of a C-decorated tree naturally correspond to the vertices of the associated unicellular map, in such a way that the respective underlying graphs are the same.

Proof.

The proof is a simple induction on , whereas is fixed. The case is obvious, as there are different -permutations of size and genus , corresponding to the ways of giving signs to the identity permutation. Let . The induction hypothesis ensures that for each , , where the underlying graphs (taking marked vertices vertices into account) of corresponding objects are the same. Hence, by Propositions 1 and 4, we have , where the underlying graphs of corresponding objects are the same. Finally, one can extract from this -to- correspondence a -to- correspondence (think of extracting a perfect matching from a -regular bipartite graph, which is possible according to Hall’s marriage theorem). And obviously the extracted -to- correspondence, which realizes , also preserves the underlying graphs. ∎

2.5. A fractional, or stochastic, formulation

Even if this does not hinder enumerative applications to be detailed in the next section, we do not know of an effective (polynomial-time) way to implement the bijection of Theorem 5; indeed the last step of the proof is to extract a perfect matching from a -regular bipartite graph whose size is exponential in .

What can be done effectively (in time complexity ) is a fractional formulation of the bijection. For a finite set , let be the set of linear combinations of the form , where the are seen as independent formal vectors, and the coefficients are in . Let be the subset of linear combinations where the coefficients are nonnegative and add up to . Denote by the vector . For two finite sets and , a fractional mapping from to is a linear mapping from to such that the image of each is in ; the set of elements of whose coefficients in are strictly positive is called the image-support of . Note that identifies to a probability distribution on ; a “call to ” is meant as picking up under this distribution. A fractional mapping is bijective if is mapped to , and is deterministic if each is mapped to some . Note that, if there is a fractional bijection from to , then (indeed in that case the matrix of is bistochastic).

One can now formulate by induction on the genus an effective (the cost of a call is ) fractional bijection from to , and similarly from to . The crucial property is that, for and , finite sets, if there is a fractional bijection from to then one can effectively derive from it a fractional bijection from to : for , just define as , where are the representatives of in , and where is the projection from to . In other words a call to consists in picking up a representative of in uniformly at random and then calling . Hence by induction on , Propositions 1 and 4 (where the stated combinatorial isomorphisms are effective) ensure that there is an effective fractional bijection from to and similarly from to , such that if is in the image-support of then the underlying graphs of and are the same.

Note that, given an effective fractional bijection between two sets and , and a uniform random sampling algorithm on the set , one obtains immediately a uniform random sampling algorithm for the set . In the next section, we will use our bijection to prove several enumerative formulas for unicellular maps, starting from elementary results on the enumeration of trees or permutations. In all cases, we will also be granted with a uniform random sampling algorithm for the corresponding unicellular maps, though we will not emphasize this point in the rest of the paper.

3. Counting formulas for unicellular maps

It is quite clear that C-decorated trees are much simpler combinatorial objects than unicellular maps. In this section, we use them to give bijective proofs of several known formulas concerning unicellular maps. We focus on the Lehman-Walsh and the Goupil-Schaeffer formulas, and the Harer-Zagier recurrence, of which bijective proofs were long-awaited. We also give new bijective proofs of several summation formulas (for which different bijective proofs are already known): the Harer-Zagier summation formula and a refinement of it, a formula due to Jackson for bipartite maps and its refinement due to A. Morales and E. Vassilieva. We finally consider an identity involving covered maps, obtained originally from a difficult bijection by the first author and O. Bernardi and which can be explained easily thanks to our new bijection. We insist on the fact that all these proofs are elementary consequences of our main bijection (Theorem 5).

3.1. Two immediate corollaries

The set is the product of two sets that are easy to count. Precisely, let and . Recall that , where is the -th Catalan number. Therefore Theorem 5 gives

One gets easily a closed form for (by summing over all possible cycle types) and an explicit formula for the generating series, thereby recovering two classical results for the enumeration of unicellular maps.

Every partition of in odd parts writes as for some partition of . The number of permutations of elements with cycle-type equal to is classically given by

 aγ(n+1)=(n+1)!(n+1−2g−ℓ)!∏imi!(2i+1)mi,

and the number of C-permutations with this cycle-type is just (since each cycle has possible signs). Hence, we get the equality

 cg(n+1)=2n+1−2g∑γ⊢gaγ(n+1).

We thus recover:

Proposition 6 (Walsh and Lehman [32]).

The number is given by

 ϵg(n)=(2n)!n!(n+1−2g)!22g∑γ⊢g(n+1−2g)ℓ∏imi!(2i+1)mi,

where , is the number of parts of , and is the number of parts of length in .

Define the exponential generating function

 C(x,y):=∑n,g1(n+1)!cg(n+1)yn+1xn+1−2g

of C-permutations where marks the number of elements, which are labelled, and marks the number of cycles. Since a C-permutation is a set of signed cycles of odd lengths, is given by

 C(x,y)=exp(2x∑k≥0y2k+12k+1)−1.

Indeed the sum in the parenthesis is the generating function of cycles of odd lengths, the factor is for the signs of cycles, the means that we take a set of such signed cycles, and the means that the set is non-empty (see e.g. [15, Part A] for a general presentation of the methodology to translate classical combinatorial set operations into generating function expressions, in particular page 120 for the application to permutations seen as sets of cycles). Since , the expression simplifies to

 C(x,y)=exp(xlog(1+y1−y))−1=(1+y1−y)x−1.

Since and for , we recover:

Proposition 7 (Harer-Zagier series formula [19, 22]).

The generating function

is given by

 E(x,y)=(1+y1−y)x.

3.2. Harer-Zagier recurrence formula

Elementary algebraic manipulations on the expression of yield a very simple recurrence satisfied by , known as the Harer-Zagier recurrence formula (stated in Proposition 10 hereafter). We now show that the model of C-decorated trees makes it possible to derive this recurrence directly from a combinatorial isomorphism, that generalizes Rémy’s beautiful bijection [28] formulated on plane trees.

It is convenient here to consider C-decorated trees as unlabelled structures: precisely we see a C-decorated tree as a plane tree where the vertices are partitioned into parts of odd size, where each part carries a sign or , and such that the vertices in each part are cyclically ordered (the C-permutation can be recovered by numbering the vertices of the tree according to a left-to-right depth-first traversal), think of Figure 1(c) where the labels have been taken out. We take here the convention that a plane tree with edges has corners, considering that the sector of the root has two corners, one on each side of the root.

We denote by the set of plane trees with edges, and by (resp. ) the set of plane trees with edges where a vertex (resp. a corner) is marked. Rémy’s procedure, shown in Figure 4, realizes the isomorphism , or equivalently

 (2) (n+1)P(n)≃2(2n−1)P(n−1).

Let be the set of C-decorated trees from where a vertex is marked. Let (resp. ) be the subset of objects in where the signed cycle containing the marked vertex has length (resp. length greater than ). Let , with . If , record the sign of the -cycle containing and then apply Rémy’s procedure to the plane tree with respect to (so as to delete ). This reduction, which does not change the genus, yields . If , let be the cycle containing the marked vertex ; is of the form for some . Move and out of (the successor of becomes the former successor of ). Then apply Rémy’s procedure twice, firstly with respect to (on a plane tree with edges), secondly with respect to (on a plane tree with edges). This reduction, which decreases the genus by , yields , hence . Since and , we finally obtain the isomorphism

 (3) (n+1)Tg(n)≃4(2n−1)Tg(n−1)+4(n−1)(2n−1)(2n−3)Tg−1(n−2),

which holds for any and (with the convention if or is negative). Since , we recover:

Proposition 8 (Harer-Zagier recurrence formula [19, 22]).

The coefficients satisfy the following recurrence relation valid for any and (with and if or ):

 (n+1)ϵg(n)=2(2n−1)ϵg(n−1)+(n−1)(2n−1)(2n−3)ϵg−1(n−2).

To the best of our knowledge this is the first proof of the Harer-Zagier recurrence formula that directly follows from a combinatorial isomorphism. The isomorphism (3) also provides a natural extension to arbitrary genus of Rémy’s isomorphism (2).

3.3. Refined enumeration of bipartite unicellular maps

In this paragraph, we explain how to recover a formula due to A. Goupil and G. Schaeffer [18, Theorem 2.1] from our bijection. Let us first give a few definitions. A graph is bipartite if its vertices can be colored in black and white such that each edge connects a black and a white vertex. If the graph has a root-vertex , then is required to be black; thus, if the graph is also connected, then such a bicoloration of the vertices is unique. From now on, a connected bipartite graph with a root-vertex is assumed to be endowed with this canonical bicoloration.

The degree distribution of a map/graph is the sequence of the degrees of its vertices taken in decreasing order (it is a partition of , where is the number of edges). If we consider a bipartite map/graph, we can consider separately the white vertex degree distribution and the black vertex degree distribution, which are two partitions of .

Let be positive integers such that is even. Fix two partitions , of of respective lengths and . We call the number of bipartite unicellular maps, with white (resp. black) vertex degree distribution (resp. ). The corresponding genus is .

The purpose of this paragraph is to compute . It will be convenient to change a little bit the formulation of the problem and to consider labelled maps instead of the usual non-labelled maps: a labelled map is a map whose vertices are labelled with integers . If the map is bipartite, we require instead that the white and black vertices are labelled separately (with respective labels and ). The degree distribution(s) of a labelled map (resp. bipartite labelled map) with edges can be seen as a composition of (resp. two compositions of ). For and two compositions of , we denote by the number of labelled bipartite unicellular maps with white (resp. black) vertex degree distribution (resp. ). The link between and is straightforward: where and are the sorted versions of and . We now recover the following formula:

Proposition 9 (Goupil and Schaeffer [18, Theorem 2.1]).
 (4) BiL(I,J)=2−2g⋅n⋅∑g1+g2=g(ℓ+2g1−1)!(m+2g2−1)!∑p1+⋯+pℓ=g1q1+⋯+qm=g2ℓ∏r=112pr+1(ir−12pr)m∏r=112qr+1(jr−12qr).
Proof.

For the formula is simply

 (5) BiL(I,J)=n(ℓ−1)!(m−1)!,

which can easily be established by a bivariate version of the cycle lemma, see also [16, Theorem 2.2]. (Note that, in that case, the cardinality only depends on the lengths of and .)

We now prove the formula for arbitrary . Consider some lists and of nonnegative integers with total sum : let and . We say that a composition refines along if is of the form with for all between and . Clearly, there are such compositions . One defines similarly a composition refining along .

Consider now the set of labelled bipartite plane trees of vertex degree distributions and , where (resp. ) refines (resp. ) along (resp. . By (5), there are trees for each pair , so in total, with , , and fixed, the number of such trees is:

 (6) n⋅(ℓ+2g1−1)!(m+2g2−1)!ℓ∏r=1(ir−12pr)m∏r=1(jr−12qr).

As the parts of (resp. ) are naturally indexed by pairs of integers, we can see these trees as labelled by the set There is a canonical permutation of the vertices of the trees with cycles of odd sizes and which preserves the bicoloration: just send to (resp. to ), where is meant modulo (resp. ). If we additionally put a sign on each cycle, we get a C-decorated tree (with labelled cycles) that corresponds to a labelled bipartite map with white (resp. black) vertex degree distribution (resp. ). Conversely, to recover a labelled bipartite plane tree from such a C-decorated tree, one has to choose in each cycle which vertex gets the label or , and one has to forget the signs of the cycles. This represents a factor .

Multiplying (6) by the above factor, and summing over all possible sequences and of total sum , we conclude that the number of C-decorated trees associated with labelled bipartite unicellular maps of white (resp. black) vertex degree distribution (resp. ), is equal to times the right-hand side of (4). By Theorem 5, this number is also equal to . This ends the proof of Proposition 9. ∎

This is the first combinatorial proof of (4) (the proof by Goupil and Schaeffer involves representation theory of the symmetric group). Moreover, the authors of [18] found surprising that “the two partitions contribute independently to the genus”. With our approach, this is very natural, since the cycles are carried independently by white and black vertices.

Remark 1.

If we set , we find the number of monochromatic maps with edges and vertex-degree distribution . This explains why we did not consider separately the monochromatic and bipartite cases (as we do in the next section).

3.4. Counting colored maps

In this paragraph, we deal with what was presented in the introduction as the first type of formulas. These formulas give an expression for a certain sum of coefficients counting unicellular maps, the expressions being usually simpler than those for the counting coefficients taken separately (like the Goupil-Schaeffer’s formula). These sums can typically be seen as counting formulas for colored unicellular maps (where the control is on the number of colors, which gives indirect access to the genus).

3.4.1. A summation formula for unicellular maps.

We begin with Harer-Zagier’s summation formula [19, 22] (which can also be very easily derived from the expression of ). In contrast to the formulas presented so far, this one has already been given combinatorial proofs [23, 17, 2] using different bijective constructions, but we want to insist on the fact that our construction gives bijective proofs for all the formulas in a unified way.

Proposition 10 (Harer-Zagier summation formula [19, 22]).

Let be the number of unicellular maps with edges and vertices. Then for

 ∑vA(v;n)xv=(2n−1)!!∑r≥12r−1(nr−1)(xr).
Proof.

To be comprehensive, let us begin by explaining the well-known combinatorial reformulation of this formula. Let be the number of unicellular maps with edges, each vertex having a color in , and each color in being used at least once. Then

 (7) ∑vA(v;n)xv=∑r≥1Ar(n)(xr).

Indeed, both sides count the number of pairs , where is a unicellular map with edges and is a mapping from the vertex set of to a given set of size . For the left-hand side, this is clear: for a given map, there are such mappings, where is the number of vertices of the map. But we can count these pairs in another way. Let us consider the pairs for which the image set of is a given set . If is the size of , such a pair is the same thing as a unicellular map colored with colors in and each color being used at least once. Therefore, for a fixed , one has such pairs . As there are sets of size , there are in total

 ∑r≥1Ar(n)(xr)

pairs , which proves identity (7).

Thus, it suffices to prove that . Our main bijection sends unicellular maps colored with colors in (each color being used at least once) onto C-decorated trees with edges, where each (signed) cycle has a color in , and such that each color in is used by at least one cycle. Each of the colors yields a (non-empty) C-permutation, which can be represented as a signed sequence, according to Lemma 3. Then one can encode these signed sequences by the triple where is their concatenation (it is a sequence of length ), where is the -tuple giving their signs and where is the subset of elements among the elements between positions and in , that indicates the starting elements of the sequences . For instance if and if the signed sequences corresponding respectively to colors are , , and , then the concatenated sequence is , together with the signs and the two selected elements . It is clear that this correspondence is bijective. Hence the number of such C-decorated trees is , and by Theorem 5,

3.4.2. A summation formula for bipartite unicellular maps.

By Theorem 5, a C-decorated tree associated to a bipartite unicellular map is a bipartite plane tree such that each signed cycle must contain only white (resp. black) vertices. Recall that the vertices carry distinct labels from to (the ordering follows by convention a left-to-right depth-first traversal, see Figure 1(c)). Without loss of information the black vertices (resp. white vertices) can be relabelled from to (resp. from to ) in the order-preserving way; we take here this convention for labelling the vertices of such a C-decorated tree. We now recover the following summation formula due to Jackson (different bijective proofs have been given in [29] and in [2]):

Proposition 11 (Jackson’s summation formula [21]).

Let be the number of bipartite unicellular maps with edges, black vertices and white vertices. Then for

 ∑v,wB(v,w;n)yvzw=n!∑r,s≥1(n−1r−1,s−1)(yr)(zs).
Proof.

As for the Harer-Zagier formula, there is a well-known combinatorial reformulation of this statement. Namely, it suffices to prove that, for , the number of bipartite unicellular maps with edges, each black (resp. white) vertex having a so-called b-color in (resp. a so-called w-color in ), such that each b-color in (resp. w-color in ) is used at least once, is given by . For such that , consider a bipartite C-decorated tree with edges, black vertices, white vertices, where each black (resp. white) signed cycle has a b-color in (resp. a w-color in ), and each b-color in (resp. w-color in ) is used at least once. By the same argument as in Proposition 10, the C-permutation and b-colors on black vertices can be encoded by a sequence of length of distinct integers in , together with a sequence of signs and a subset of elements among the elements at positions from to in . And the C-permutation and w-colors on white vertices can be encoded by a sequence of length of distinct integers in , together with a sequence of signs and a subset of elements among the elements at positions from to in . Hence there are such C-decorated trees, where (called the Narayana number) is the number of bipartite plane trees with edges, black vertices and white vertices, given by . By Theorem 5,

 Br,s(n) = 2−n−12r+s∑i+j=n+1Nar(i,j;n)i!j!(i−1r−1)(j−1s−1) = n!(n−1)!2r+s−n−1(r−1)!(s−1)!∑i+j=n+1i≥r,j≥s1(i−r)!(j−s)!.

But we have

 ∑i+j=n+11(i−r)!(j−s)!=∑i+j=n+1−r−s1i!j!=2n+1−r−s(n+1−r−s)!.

Hence . ∎

3.4.3. A refinement of the Harer-Zagier summation formula

The proof method above can be used to keep track of the vertex degree distribution in the Harer-Zagier formula. Before stating the resulting formula, let us mention that other methods can also keep track of this statistics, for instance the bijective approach developed in [2]222 Proposition 12 can also be deduced from a formula of A. Morales and E. Vassilieva (Proposition 14 below) using [5, Lemma 9].. Thus, although the formula had not yet been stated explicitly in the literature (as far as we know), all the elements needed to prove it were already there333This formula was known to an anonymous referee, who suggested we include its proof in the present paper.. Of course, the proof presented here is new and fits in our unified framework.

Proposition 12.

Let and be the monomial and power sum bases of the ring of symmetric functions and be an infinite set of variables. We denote by the number of unicellular maps with degree distribution . Then, for any integer , one has:

 ∑λ⊢2nAi(λ)pλ(x)=∑ρ⊢2nn(2n−ℓ(ρ))!(n−ℓ(ρ)+1)!2ℓ(ρ)−nmρ(x).

Let us make three remarks on this statement. First, together with the trivial fact that if is a partition of an odd number, it entirely determines the numbers (as power sums form a basis of the ring of symmetric functions). Second, it implies the Harer-Zagier summation formula (which can be recovered by setting with exactly times the value ). Third, it admits an equivalent combinatorial formulation, that we shall present now.

As in the previous subsection, we shall consider colored maps, that is maps whose vertices are colored with numbers from to , each color being used at least one. For such a map, one can consider its colored vertex degree distribution: by definition, it is the composition such that is the sum of the degrees of the vertices of color .

We denote by the number of colored maps with colored vertex degree distribution . Then, using the tools of [24, Section 2], one can easily show that Proposition 12 is equivalent to the following statement, that we can prove using our main bijection.

Proposition 13.

For any composition of of length ,

 AiC(I)=n(2n−r)!(n−r+1)!2r−n.
Proof.

We shall first consider the case where has length . This means that we count rooted unicellular maps with edges and vertices of colors. But a unicellular map is necessarily connected and, hence, has at most vertices. Therefore, we are counting rooted unicellular maps with labelled vertices, that is, rooted plane trees with labelled vertices of prescribed degrees. In that case, one can show that by several methods (e.g., the cycle lemma, or Pitman’s aggregation process [20], or the more recent method by Bernardi and Morales [6]). We give here a short proof by induction.Note that an unrooted vertex-labelled plane tree cannot have any symmetry and thus can always be rooted in ways. So we shall rather compute the number of unrooted vertex-labelled plane trees with degree distribution .

We shall prove by induction that .

For , the only possibility is and there is only one tree with such degree distribution: . Thus .

Let be a composition of of length . This composition must contain a part equal to and without loss of generality ( is invariant by permutation of the parts of ), we may assume that . This means that we are counting trees , in which the vertex labelled is a leaf. Denote by the label of the vertex to which this leaf is attached. Then, removing the leaf from , we obtained a tree of degree distribution