A bijection for covered maps, or a shortcut between Harer-Zagier's and Jackson's formulas
We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus ) with a marked unicellular spanning submap (which can have any genus in ). Our main result is a bijection between covered maps with edges and genus and pairs made of a plane tree with edges and a unicellular bipartite map of genus with edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in .
Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps.
We also show that the bijection of Bouttier, Di Francesco and Guitter  (which generalizes a previous bijection by Schaeffer ) between bipartite maps and so-called well-labelled mobiles can be obtained as a special case of our bijection.
We consider maps on orientable surfaces of arbitrary genus. A map is called unicellular if it has a single face. For instance, the unicellular maps of genus 0 are the plane trees. A covered map is a map together with a marked unicellular spanning submap. A map of genus can have spanning submaps of any genus in . In particular, a tree-rooted map (map with a marked spanning tree) is a covered map since the spanning trees are the unicellular spanning submaps of genus 0. A covered map of genus 2 with a unicellular spanning submap of genus 1 is represented in Figure 1(a).
Our main result is a bijection (denoted by ) between covered maps of genus with edges, and pairs made of a plane tree with edges and a bipartite unicellular map of genus with edges. If the covered map has vertices and faces, then the bipartite unicellular map has white vertices and black vertices. The bijection is represented in Figure 1. In the planar case , the bijection coincides with a construction by the first author  between planar tree-rooted maps with edges and pairs of plane trees with and edges respectively
Before discussing the bijection further, let us give another equivalent way of defining covered maps. We start with the planar case which is simpler. Let be a planar map and let be its dual. Then, for any spanning tree of , the dual of the edges not in form a spanning tree of . In other words, the dual of a planar tree-rooted map is a planar tree-rooted map. Pushing this observation further, Mullin showed in  that a tree-rooted map can be encoded by a shuffle of two trees (one representing the spanning tree on primal map , the other representing the spanning tree on the dual map ), or more precisely as a shuffle of two parenthesis systems encoding these trees. Covered maps enjoy a similar property: the dual of a covered map is a covered map. Using this observation, it is not hard to see that covered maps can be encoded by shuffles of two unicellular maps (see Section 3).
We emphasize that our bijection is not the encoding of covered maps as shuffles of two unicellular maps: the image by is a pair of unicellular maps of a fixed size, and not a shuffle. As a matter of fact, comparing the enumerative formulas given by the bijection with the formulas given by the shuffle approach yields the equivalence between formulas by Harer and Zagier , and by Jackson . As a warm up, let us consider the total number of covered maps with edges (and arbitrary genus). The bijection implies
where is the Catalan number counting rooted plane trees with edges, and is the total number of bipartite unicellular maps with edges. On the other hand, the shuffle approach implies
where is the total number of unicellular maps with edges (and the term accounts for the shuffling). Here, the relation between (1) and (2) is merely an application of the Chu-Vandermonde identity, but things get more interesting when one considers refinements of these equations. First, by the bijection , the number of covered maps of genus with edges is
where is the number of bipartite unicellular maps of genus with edges. A further refinement can be obtained by taking into account the numbers and of vertices and faces of the covered map. We show in Section 5, that comparing the formula given by the bijection with the formula given by the shuffle approach yields
where is the number of unicellular bipartite maps with edges, white vertices and black vertices, and is the number of unicellular maps with edges and vertices. Equation (4) actually establishes the equivalence between the formula of Harer and Zagier  for unicellular maps:
and the formula of Jackson  for bipartite unicellular maps:
The original proof of (5) involves a matrix integral argument; combinatorial proofs are given in [23, 18, 3, 11]. The original proof of (6) (as well as another related formula by Adrianov ) is based on the representation theory of the symmetric group; a combinatorial proof is given in .
Let us mention a few other enumerative corollaries of the bijection .
First, plugging the identity
This formula is originally due to Zagier , and has been given a bijective proof (different from ours) by Feray and Vassilieva . We now turn to formulas concerning the number of tree-rooted maps of genus with edges. In the planar case, tree-rooted maps are the same as covered maps. Hence, (3) gives
This formula was originally proved by Mullin  (using the shuffle approach) who asked for a direct bijective proof; this was the original motivation for the planar specialization of described in  (and for a related recursive bijection due to Cori, Dulucq and Viennot ). In the case of genus , a duality argument shows that exactly half of the covered maps are tree-rooted maps, so that (3) gives
This formula was originally proved by Lehman and Walsh (using the shuffle approach), and had no direct bijective proof so far.
We now explain the relation between the bijection and some known bijections. In [5, 6], two “master bijections” for planar maps are defined, and then specialized in various ways so as to unify and extend several known bijections (roughly speaking, by specializing the master bijections appropriately, one can obtain a bijection for any class of planar maps defined by a girth condition and a face-degree condition). The master bijections are in fact “variants” of the planar version of the bijection . Here we shall prove that the bijection generalizes the bijection obtained for bipartite planar maps by Bouttier, Di Francesco and Guitter [8, Section 2] (however, we do not recover the most general version of their bijection [8, Section 3-4]), as well as its generalization to higher genus surfaces by Chapuy, Marcus and Schaeffer [13, 12]. These bijections (which themselves generalize a previous bijection by Schaeffer ) are of fundamental importance for studying the metric properties of random maps [14, 7, 10, 9, 30] and for defining and analyzing their continuous limit, the Brownian map [29, 24, 26, 25].
The paper is organized as follows. In Section 2, we recall some definitions about maps. In Section 3, we show that covered maps can be encoded by shuffles of unicellular maps. In Section 4, we define the bijection between covered maps with edges and pairs made of a plane tree with edges and a bipartite unicellular map with edges. In Section 5, we explore the enumerative implications of the bijection . In Section 6, we prove the bijectivity of . In Section 7, we give three equivalent ways of describing the image of (the pairs made of a plane tree and a bipartite unicellular map) and use one of these descriptions in order to show that the bijections for bipartite maps described in [8, 13] are specializations of . Lastly, in Section 8, we study the properties of the bijection with respect to duality.
Maps. Maps can either be defined topologically (as graphs embedded in surfaces) or combinatorially (in terms of permutations). We shall prove our results using the combinatorial definition, but resort to the topological interpretation in order to convey intuitions.
We start with the topological definition of maps. Here, surfaces are -dimensional, oriented, compact and without boundaries. A map is a connected graph embedded in surface, considered up to orientation preserving homeomorphism. By embedded, one means drawn on the surface in such a way that the edges do not intersect and the faces (connected components of the complement of the graph) are simply connected. Loops and multiple edges are allowed. The genus of the map is the genus of the underlying surface and its size is its number of edges. A planar map is a map of genus . A map is unicellular if it has a single face. For instance, the planar unicellular maps are the plane trees. A map is bipartite if vertices can be colored in black and white in such a way that every edge join a white vertex to a black vertex. We denote by the genus of a map and by , , respectively its number of vertices, faces and edges. These quantities are related by the Euler formula:
By removing the midpoint of an edge, one obtains two half-edges. Two consecutive half-edges around a vertex define a corner. A map is rooted if one half-edge is distinguished as the root. The vertex incident to the root is called root-vertex. In figures, the rooting will be indicated by an arrow pointing into the root-corner, that is, the corner following the root in clockwise order around the root-vertex. For instance, the root of the map in Figure 2 is the half-edge .
Maps can also be defined in terms of permutations acting on half-edges.
To obtain this equivalence, observe first that the embedding of a graph in a surface defines a cyclic order (the counterclockwise order) of the half-edges around each vertex. This gives in fact a one-to-one correspondence between maps and connected graphs together
with a cyclic order of the half-edges around each vertex
(see e.g. ). Equivalently, a map can be defined as a triple
where is a finite set whose elements are called the
half-edges, is an involution of without fixed point, and
is a permutation of such that the group generated by and
acts transitively on (here we follow the notations in ). This must be understood as follows: each cycle of describes the counterclockwise order of the half-edges around one vertex of the map, and each cycle of describes an edge, that is, a pair of two half-edges; see Figure 2 for an example. The transitivity assumption simply translates the fact that the graph is connected.
For a map , the permutation is called vertex-permutation, the permutation is called edge-permutation and the permutation is called face-permutation. The cycles of , , are called vertices, edges and faces. Observe that the cycles of are indeed in bijection with the faces of the map in its topological interpretation. Hence, the genus of can be deduced from the number of cycles of , and by the Euler relation. We say that a half-edge is incident to a vertex or a face if this edge belongs to the corresponding cycle. Again, a map is rooted if one of the half-edges is distinguished as the root; the incident vertex and face are called root-vertex and root-face.
The correspondence between topological and combinatorial maps is one-to-one if combinatorial maps are considered up to isomorphism (or, relabelling). That is, two maps and are considered the same if there exists a bijection such that and (for rooted maps, we ask furthermore that ). In this article all maps will be rooted, and considered up to isomorphism.
We call pseudo map a triple such that is a fixed-point free involution, but where the transitivity assumption (i.e. connectivity assumption) is not required. This can be seen as a union of maps and we still call the face-permutation, as its cycles are indeed in correspondence with the faces of the union of maps. Lastly, we consider the case where the set of half-edges is empty as a special case of rooted unicellular map (corresponding to the planar map with one vertex and no edge) called empty map.
Submaps, covered maps and motion functions.
For a permutation on a set , we call restriction of to a set and denote by the permutation of whose cycles are obtained from the cycles of by erasing the elements not in . Observe that so that we shall not use parenthesis anymore in these notations. It is sometime convenient to consider the restriction as a permutation on the whole set acting as the identity on ; we shall mention this abuse of notations whenever necessary.
A spanned map is a map with a marked subset of edges.
In terms of permutations, a spanned map is a pair , where is a subset of half-edges stable by the edge-permutation . The submap defined by , denoted , is the pseudo map , where is the vertex-permutation of . We underline that the face-permutation of the pseudo-map is not equal to . Observe also that the genus of can be less than the genus of . For example, Figure 1(a) represents a submap of genus 1 of a map of genus 2.
A submap is connecting if it is a map containing every vertex of , that is, contains a half-edge in every vertex of (except if has a single vertex, where we authorize to be empty) and , act transitively on . The submap represented in Figure 3 (right) is a map but is not connecting. A covered map is a spanned map such that the submap is a connecting unicellular map. A tree-rooted map is a spanned map such that the submap is a spanning tree, that is, a connecting plane tree.
The motion function of the spanned map is the mapping defined on by if is in and otherwise. Observe that the stability of by implies that the motion function is a permutation of since its inverse is given by if is in and otherwise. Observe also that, given the map , the set can be recovered from the motion function . Topologically, the motion function is the permutation describing the tour of the faces of the connected components of the submap in counterclockwise direction: we follow the border of the edges of the submap and cross the edges not in . See Figure 3 for an example.
An orientation of a map is a partition such that the involution maps the set of ingoing half-edges to the set of outgoing half-edges. The pair is an oriented map.
A directed path is a sequence of distinct ingoing half-edges such that , are incident to the same vertex (are in the same cycles of ) for . A directed cycle is a directed path such that and are incident to the same vertex. The half-edge is called the extremity of the directed path. An orientation is root-connected if any ingoing half-edge is the extremity of a directed path such that is incident to the root-vertex of .
Duality The dual map of a map is the map where is the face-permutation of . The root of the dual map is equal to the root of . Observe that the genus of a map and of its dual are equal (by Euler relation) and that . Topologically, the dual map is obtained by the following two steps process: see Figure 4.
In each face of , draw a vertex of . For each edge of separating faces and (which can be equal), draw the dual edge of going from to across .
Flip the drawing of , that is, inverse the orientation of the surface.
We now define duals of spanned maps and oriented maps. Given a subset , we denote . The dual of a spanned map is the spanned map ; see Figure 4. We also say that and are dual submaps. Observe that the motion functions of a spanned map and of its dual are equal. The dual of the oriented map is . Graphically, this orientation is obtained by applying the following rule at step 1: the dual-edge of an edge is oriented from the left of to the right of ; see Figure 14. Observe that duality is involutive on maps, spanned maps and oriented maps.
3. Covered maps as shuffles of unicellular maps.
In this Section, we establish some preliminary results about covered maps. In particular we prove that covered maps are stable by duality and explicit their encoding as shuffles of two unicellular maps. Our first result should come as no surprise: it simply states that a spanned map is a covered map if and only if turning around the submap (that is following the border of its edges) starting from the root allows one to visit every half-edge of .
Let be a spanned map, and let , and be the vertex-, edge-, and face-permutations of . The motion function satisfies and . That is, the restriction is the face permutation of the pseudo map , while the restriction is the face-permutation of the dual pseudo map .
In particular a spanned map is a covered map if and only if its motion function is a cyclic permutation.
Proposition 3.1 is a consequence of the following lemma:
Let and be two permutations on the set , let be a subset of , and let be the mapping defined by:
Then is a permutation if and only if is stable by . Moreover in that case we have:
Proof of Lemma 3.2.
First, if is stable by then the inverse of is given by if and is (note that is equivalent to since is stable by ). Conversely, if there exists such that , then and is not a permutation. This proves the first claim.
For the second claim, let and . By definition of the restriction we have , and there exist such that , for , and . Moreover by definition of we have and for . Now, since is stable by , we have , which implies that by definition of the restriction. ∎
Proof of Proposition 3.1.
The fact comes from Lemma 3.2, and the relation follows from the preceding point by duality (since the motion functions of a spanned map and its dual are equal).
Now let be a covered map. Since is connecting, each cycle of the motion function contains an element of . Hence, the number of cycles of and is the same. Moreover, by Lemma 3.2, is the face-permutation of . Since is unicellular, is cyclic and is also cyclic.
Conversely, suppose that the motion function is cyclic. In this case, the pseudo map has a face-permutation which is cyclic by Lemma 3.2. Hence it is a unicellular map. ∎
Proposition 3.1 immediately gives the following corollary concerning duality.
If a spanned map is a covered map, then the dual spanned map is also a covered map. Moreover the genus of is the sum of the genera of the unicellular maps and :
The fact that is a covered map is an immediate consequence of Proposition 3.1 since the motion function of a submap and of its dual are always equal. The fact that the genus adds up is obtained by writing the Euler relation for the maps , and . ∎
Let be a covered map. By Proposition 3.1, the restrictions and of the motion function correspond respectively to the face-permutations of the unicellular maps and . This inclines to say, somewhat vaguely, that the covered map is a shuffle of the unicellular maps and . Making this statement precise requires introducing codes of unicellular maps and covered maps.
A unicellular code on the alphabet is a word on such that every letter of appears exactly once, and for all , the letter appears before and before . Let be a unicellular map with edges. By definition, the face-permutation is cyclic. Hence, there exists a unique way of relabelling the half-edges on the set in such a way that for all and , where is the root and is a unicellular code. We call the code of the unicellular map .
Topologically, the code of a unicellular map is obtained by turning around the face of the map in counterclockwise direction starting from the root and writing when we discover the th edge and writing when we see this edge for the second time. For instance, the code of the unicellular map in Figure 2 is . We also mention that the unicellular map is a plane tree if and only if its code does not contain a subword of the form . In this special case, replacing all the letters of the code by the letter and all the letters by the letter results in no loss of information. One thereby obtains the classical bijection between plane trees and parenthesis systems on .
Lemma 3.4 (Folklore).
The mapping which associates its code to a unicellular map is a bijection between unicellular map with edges and unicellular code on the alphabet .
The mapping is injective since the root and the edge-permutation and vertex-permutation can be recovered from the code. It is also surjective since starting from any code one obtains a pair of permutation which indeed gives a unicellular map (the only non-obvious property is the transitivity condition, but this is granted by the fact the face-permutation is cyclic). ∎
A word on (where ) is a code-shuffle if the subwords and made of the letters in and respectively are unicellular codes on and . Let be a covered map, where and let , . By Lemma 3.1, the motion function is cyclic. Hence, there exists a unique way of relabelling the half-edges on the set in such a way that , , for all , for all , and , where is the root of and is a code-shuffle. We call the code of the covered map .
Topologically, the code of a covered map is obtained by turning around the submap in counterclockwise direction starting from the root and writing (resp. ) when we discover the th edge in (resp. ) and writing (resp. ) when we see this edge for the second time. For instance, the code of the covered map in Figure 7(a) is . We now state the main result of this preliminary section.
The mapping which associates its code to a covered map is a bijection between covered maps with edges and code-shuffles of length . Moreover, if is the code of the covered map , then is the code of the unicellular map (on the alphabet ) and is the code of the dual unicellular map (on the alphabet ).
To see that is injective, observe first that the code-shuffle allows to recover the root of the map , the subset , the edge-permutation and the motion function . From this, the vertex-permutation is deduced by if and otherwise. We now prove that is surjective. For this, it is sufficient to prove that starting from any shuffle-code, the pair defined as above is a covered map. First note that the permutations and clearly act transitively on since is cyclic, hence is a map. Now, the fact that is a a covered map is a consequence of Lemma 3.1 since is the motion function of and is cyclic.
We now prove the second statement. Let and . By definition of restrictions, and . Moreover, by Proposition 3.1, these restrictions and correspond respectively to the face-permutations of and . Recall also that the root of is , where is the root of and is the least integer such that . Equivalently, where is the least integer such that , hence . Similarly, the root of is where is the least integer such that , or equivalently where is the least integers such that , hence . Thus, the words and are the codes of the unicellular maps and respectively. ∎
We now explore the enumerative consequence of Proposition 3.5. Let be the number of unicellular maps of genus with edges. Let (resp. ) be the number of covered maps such that the unicellular maps and have respectively and edges (resp. a total of edges) and genus and . Since there are ways of shuffling unicellular codes of length and , Proposition 3.5 gives
An alternative equation (used in Section 5) is obtained by fixing the number of vertices of and instead of their genus. Let be the number of unicellular maps with vertices and edges ( by Euler relation and this number is 0 if is odd). Let also be the number of covered maps with vertices, faces and edges (hence with genus ). Proposition 3.5 gives
where is the th Catalan number. In , Mullin proved Equation (7) by applying the Chu-Vandermonde identity to (12) (in the case ). Similarly, in , Lehman and Walsh proved Equation (8) by applying the Chu-Vandermonde identity to (12) (in the case ). In , Bender et al. used the asymptotic formula
Applying the same techniques as Bender et al. to Equation (10) gives the asymptotic number of covered maps:
In particular, the the total number of covered maps of genus with edges satisfies:
Hence the proportion of tree-rooted maps among covered maps of genus tends to when the size goes to infinity. We have no simple combinatorial interpretation of this fact.
This concludes our preliminary exploration of covered maps. We now leave the world of shuffles and concentrate on the main subject of this paper, that is, the bijection between covered maps and pairs made of a tree and a unicellular bipartite map.
4. The bijection.
This section contains our main result, that is, the description of the bijection between covered maps and pairs made of a tree and a bipartite unicellular map called the mobile.
Let be a covered map. The bijection consists of two steps. At the first step, the submap is used to define an orientation of ; see Figure 7. At the second step of the bijection, which we call unfolding, the vertices of the map are split into several vertices (the rule for the splitting is given in terms of the orientation ; see Figure 6).
The map obtained after these splits is a plane tree , and the information about the splitting process can be encoded into a bipartite unicellular map . The tree and the mobile are represented in Figure 9.
We now describe the two steps of the bijection in more details.
Step 1: Orientation . The orientation step is represented in Figure 7. One starts with a covered map and obtains an oriented map . Topologically, the orientation is obtained by turning around the submap (in counterclockwise direction starting from the root) and orient each edge of according to the following rule:
each edge in is oriented in the direction it is followed for the first time during the tour,
each edge not in is oriented in such a way that the ingoing half-edge is crossed before the outgoing half-edge during the tour.
Let us now make definitions precise in terms of the combinatorial definition of maps. Let be a covered map, let be its root, and let be its motion function. Recall from Proposition 3.1 that is a cyclic permutation on the set of half-edges. Therefore, one obtains a total order , named appearance order, on the set by setting . Topologically, the appearance order is the order in which half-edges of appear when turning around the spanning submap in counterclockwise order starting from the root. For instance, the order obtained for the spanning submap in Figure 7(a) is . We now define the oriented map which is represented in Figure 7(b).
Let be a covered map with half-edge set . The mapping associates to the oriented map , where the set of ingoing half-edges contains the half-edges such that and the half-edges such that (and ).
We now characterize the image of the mapping by defining left-connected orientations. Let be a map and let be an orientation. Let denote the root of . A left-path is a sequence of ingoing half-edges such that for all , there exists an integer such that and for all . In words, a left-path is a directed path starting from the arrow pointing the root-corner and such that no ingoing half-edges is incident to the left of the path. Clearly, for any ingoing half-edge , there exists at most one left-path whose extremity is is . We say that an oriented map is left-connected if every ingoing half-edge is the extremity of a left-path.
The mapping is a bijection between covered maps and left-connected maps.
Remark on the planar case: It is shown in [4, Prop. 3] that the mapping is a bijection between planar covered maps (i.e. tree-rooted maps) and planar oriented maps which are accessible (any vertex can be reach from the root-vertex by a directed path) and minimal (no directed cycle has the root-face on its right). Thus, in the planar case the left-connected orientations are the accessible minimal orientations.
Step 2: Unfolding .
The unfolding step is represented in Figures 8 and 9. One starts with a left-connected map and obtains two maps and . The map is a plane tree and the map is a bipartite unicellular map (with black and white vertices). By analogy with the paper , we call the bipartite unicellular the mobile associated with the left-connected map .
Let us start with the topological description of this step. Let be a vertex of the oriented map and let be the incident half-edges in counterclockwise order around (here it is important to consider the arrow pointing the root-corner as an ingoing half-edge). If the vertex is incident to ingoing half-edges, say , then the vertex of will be split into vertices of the tree . The splitting rule is represented in Figure 6: for , the vertex of the trees is incident to the half-edges .
Observe that the splitting of the vertex can be written conveniently in terms of permutations. Indeed, seeing the vertex as the cycle of the vertex-permutation and the vertices , …, as cycles of the vertex-permutation of the tree gives the following relation between and the product of cycles (these are both permutations on )
where is the permutation such that if and for . Hence, , where (with the convention that the restriction acts as the identity on ). The cycle of will represent one of the (white) vertices of the bipartite unicellular map . This white vertex is represented in Figure 8(a).