aa r X i v : . [ m a t h . C O ] M a r New bijective links on planar maps via orientations ´Eric Fusy
Abstract.
This article presents new bijections on planar maps. At first a bijection is established betweenbipolar orientations on planar maps and specific “transversal structures” on triangulations of the 4-gon withno separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijectionbetween rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijectionbetween rooted loopless maps and rooted triangulations, based on the observation that loopless maps andtriangulations are decomposed in a similar way into components that are respectively non-separable mapsand irreducible triangulations. This gives another bijective proof (after Wormald’s construction publishedin 1980) of the fact that rooted loopless maps with n edges are equinumerous to rooted triangulations with n inner vertices.
1. Introduction
Planar maps (connected graphs equipped with a planar embedding) are a rich source of structuralcorrespondences and enumerative results. For counting purpose, all planar maps (shortly called maps) arehere assumed to be rooted, that is, they have a marked oriented edge with the outer face on its left. Asdiscovered by Tutte in the 60’s [
Tut63 ], many families of maps (eulerian, triangulations, quadrangulations...)have strikingly simple counting sequences, which are reminiscent of formulas for families of plane trees.Tutte’s method is based on generating functions. At first a recursive decomposition of maps by root edgedeletion is translated to a recurrence on the counting sequence, and then to a functional equation satisfiedby the associated generating function. Solving this equation yields a formula for the counting sequence (bycoefficient extraction). Tutte’s method has the advantage of being quite automatic, but requires an involvedmachinery —the quadratic method— to solve equations on generating functions [
Tut73 ].By now there is a bijective approach, introduced by Schaeffer in his thesis [
Sch98 ], to prove countingformulas for many map families. The main ingredients are as follows: given a family of map M , one searchesa regular combinatorial structure —typically an orientation with simple outdegree conditions— specific tothe maps in M , which gives a way to associate with each map of M a tree with simple degree conditions.Conversely the map is recovered by performing local operations on the tree so as to close the faces oneby one. The bijection ensures that the counting sequence of M is equal to the counting sequence of theassociated tree family, which has typically a closed formula involving binomial coefficients.These enumerative techniques (Tutte’s recursive method or Schaeffer’s bijective approach) yield thusan extensive table of counting formulas for families of maps. By examining such a table, one notices thatseemingly unrelated map families are equinumerous. This asks for bijective explanations as direct as pos-sible to understand the underlying structural correspondences. On a few instances there already exist verysimple bijective constructions. Such bijections are of a type different from the above mentioned bijectiveconstructions from trees. Indeed, they are from map to map and operate directly on the embedding in asimple local way. Let us mention the classical duality construction between maps with i vertices and j facesand maps with j vertices and i faces, the radial mapping between maps with n edges and 4-regular mapswith n vertices, and the so-called trinity mapping between bipartite 3-regular maps with 2 n vertices andeulerian maps with n edges (the two latter constructions can be traced back to Tutte [ Tut63 ]). We haveobserved two further coincidences:
Mathematics Subject Classification.
Primary 05A15; Secondary 05C30.
Key words and phrases. bijections, planar maps, bipolar orientations. ´E. Fusy plane bipolar no N -patternplane bipolar posets transversal structuresorientations2-connectedmaps irreducibletriangulationsloopless maps triangulations minimal minimalglue at vertices glue at triangles no N -pattern φ φ ′ F F Figure 1.
Relations between the combinatorial structures and map families considered inthe article. Bijective links are indicated by double arrows. Below each double arrow isindicated the name of the mapping for the left-to-right arrow. (i): non-separable maps with n edges are equinumerous to so-called irreducible triangulations (trian-gulations of the 4-gon with no separating triangle) with n + 3 vertices, (ii): loopless maps with n edges are equinumerous to triangulations with n + 3 vertices.In this article, we describe new bijective constructions for proving (i) and (ii). To wit, the bijection for(i) can be seen as a direct map-to-map correspondence. But it borrows ideas from the above mentionedbijective constructions of maps from trees. At first we associate a combinatorial structure with each of thetwo map-families: plane bipolar orientations for non-separable maps and transversal structures for irreducibletriangulations (these are defined in Section 2). We show in Section 3 that these structures are closely related:to each plane bipolar orientation O is associated in a bijective way a so-called N -avoiding plane bipolar poset P = φ ( O ), which yields in turn a transversal structure X = φ ′ ( P ); the mapping Φ = φ ′ ◦ φ is bijective ontoso-called N -avoiding transversal structures.Then, the correspondence Φ specialises to a bijection, described in Section 4, between non-separablemaps and irreducible triangulations. The bijection, denoted by F , is based on the two following properties: • each non-separable map has a minimal plane bipolar orientation and each irreducible triangulationhas a minimal transversal structure; • the correspondence Φ matches the plane bipolar orientations that are minimal on their non-separable maps with the transversal structures that are minimal on their irreducible triangulations.As described in Section 5, the bijection F yields in turn a bijection, denoted F , for proving (ii).We make use of two classical decompositions: a loopless map is decomposed into a non-separable map,called the core, and a collection of components that are smaller loopless maps; whereas a triangulation isdecomposed into an irreducible triangulation, called the core, and a collection of components that are smallertriangulations. The key observation is that the two decompositions are parallel, for a convenient choice of thesize parameters. The bijection F is thus specified recursively: the cores are matched by the bijection F , andthe components are matched recursively by F . Let us mention that another bijective proof of (ii) has beendescribed by Wormald [ Wor80 ], still recursive but based on different principles (in [
Wor80 ], an isomorphismis established between the generating tree of loopless maps and the generating tree of triangulations with anadditional catalytic variable), see the discussion after Theorem 19. To sum up, the whole bijective schemeof the article is shown in Figure 1.
2. Preliminaries2.1. Planar maps. A planar map , shortly called a map, is a connected unlabelled planar graph em-bedded in the plane with no edge-crossings, the embedding being considered up to continuous deformation.Loops and multiple edges are allowed. In addition to the vertices and edges of the graph embedded, a maphas faces , which are the maximal connected areas of the plane split by the embedding. The unbounded faceis called the outer face, the other ones are called inner faces. Edges and vertices are said to be inner or outerwhether they are incident to the outer face or not. A corner of a map is a triple ( e, v, e ′ ) where v is a vertexand e and e ′ are two consecutive edges in clockwise order around v .A map is rooted by distinguishing and orienting an edge, called the root , with the condition that theroot has the outer face on its right (equivalently, it is a map with a distinguished corner incident to the outerEW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 3 a) b) W NS E
W NS E (T1):(T2): st (V):(F): Figure 2. (a) Plane bipolar orientations: local conditions and an example. (b) Transversalstructures: local conditions and an example.face). The origin of the root is called the root vertex . In this article we will consider the following familiesof (rooted) planar maps: • Loopless maps.
A loopless is a map such that each edge has two distinct extremities. Multipleedges are allowed. • non-separable maps. A map is non-separable if it is loopless and the deletion of any one vertexdoes not disconnect the map. Multiple edges are allowed. • Triangulations.
A triangulation is a map with no loop nor multiple edges and with all faces ofdegree 3. These correspond to maximal planar graphs embedded in the plane. • Irreducible triangulations.
A triangulation of the 4-gon is a map with no loop nor multiple edges,with a quadrangular outer face and triangular inner faces. An irreducible triangulation of the 4-gon,shortly called irreducible triangulation, is a triangulation of the 4-gon such that the interior of any3-cycle is a face. A bipolar orientation on a connected graph G is an acyclic orien-tation with a unique source (vertex with only outgoing edges) denoted by s and a unique sink (vertex withonly ingoing edges) denoted by t . Equivalently, the partial order induced on the vertices by the orientationhas a unique minimum and a unique maximum. Bipolar orientations constitute the natural combinatorialstructure characterising 2-connectivity. Indeed, as is well known [ dFOdMR95 ], a graph G with two markedvertices s and t admits a bipolar orientation with source s and sink t if and only if G is non-separable uponconnecting s and t by an edge.The source and the sink of the bipolar orientation are also called the poles or the special vertices . Theother vertices are said to be non-special . A plane bipolar orientation is a bipolar orientation on a planarmap M such that the source and the sink are outer vertices of M . It is convenient when considering planebipolar orientations to draw two half-lines starting respectively from s and t and reaching into the outerface. The outer face is thus split into two unbounded faces, which are called the special faces ; looking from s to t , the one on the left is called the left special face and the one on the right is called the right special face .As described in [ dFOdMR95, FFNO08 ], plane bipolar orientations have the nice property that they arecharacterised by two simple local properties, one around vertices and one around faces, see Figure 2(a): (V): Around each non-special vertex, the edges form a nonempty interval of outgoing edges and anonempty interval of ingoing edges. (F):
The contour of each inner face f is made of two oriented paths with same origin s f and sameend t f . The two special faces are each bordered by a path that goes from s to t .Given Property (V), we define the left lateral face of a non-special vertex v as the face in the corner betweenthe last ingoing edge and the first outgoing edge in clockwise order around v ; define similarly the right lateralface of v as the face in the corner between the last outgoing edge and the first ingoing edge in clockwiseorder around v . Given an inner face f , the vertices s f and t f are respectively called the source and the sink of f . The path from s f to t f that has the exterior of f on its left is called the left lateral path of f and isdenoted P ℓ ( f ); the path from s f to t f that has the exterior of f on its right is called the right lateral path ´E. Fusy of f and is denoted P r ( f ). The last edge of P ℓ ( f ) is called the topleft edge of f and the first edge of P r ( f ) iscalled the bottomright edge of f . The vertices of P ℓ ( f ) \{ s f , t f } are called left lateral vertices of f , and thevertices of P r ( f ) \{ s f , t f } are called right lateral vertices of f . The paths bordering the left special face andthe right special face are called respectively the left outer path and the right outer path .It is well known that a plane bipolar orientation O induces a partial order on the edge-set: e ≤ e ′ if andonly if there exists an oriented path passing by e before passing by e ′ . In other words ≤ is the transitiveclosure of the binary relation: e ≺ e ′ if and only if there is a vertex v such that e is ingoing at v and e ′ isoutgoing at v .Due to a classical duality relation satisfied by plane bipolar orientations [ dFOdMR95 ], there is a partialorder on the face-set and another partial order, called dual, on the edge-set. Precisely, given two faces f and f ′ (special or not), write f ≺ F f ′ if there is an edge of O with f on its left and f ′ on its right; the transitiveclosure of ≺ F is a partial order, called the left-to-right order on the faces of O . And the dual order on theedge-set, denoted ≤ ∗ , is defined as the transitive closure of the relation: e ≺ ∗ e ′ (with e and e ′ edges of O ) ifand only if there exists a face f of O such that e is in the left lateral path and e ′ is in the right lateral pathof f .Given a plane bipolar orientation with no multiple edges, a transitive edge is an edge whose two ex-tremities are connected by an oriented path of length at least 2. A plane bipolar poset is a plane bipolarorientation with at least 3 vertices, with no multiple edges, and with no transitive edges. The terminologyrefers to the fact that a plane bipolar poset is a planarly embedded Hasse diagram representing a poset.The following property is easily checked from Condition (F): Fact 1.
A plane bipolar orientation with at least 3 vertices is a plane bipolar poset if and only if the twolateral paths of each inner face have length at least 2.
An important remark to be used later is that each inner face f of a plane bipolar poset has at leastone lateral vertex on each side: the last vertex of P ℓ ( f ) \{ s f , t f } is called the topleft lateral vertex of f andthe first vertex of P r ( f ) \{ s f , t f } is called the bottomright lateral vertex of f . Given a plane bipolar poset,a N -pattern is a triple ( e , e , e ) of edges such that e and e have the same origin v , e and e havethe same end v ′ , e follows e in clockwise order around v , and e follows e in clockwise order around v ′ .The edge e is called the central edge of the N -pattern. A N -pattern is defined similarly, upon replacingclockwise by counterclockwise. Plane bipolar posets with no N -pattern are said to be N -avoiding ; theseplay an important role in the bijections to be given next. Transversal structures play a similar part for irreducible triangulationsas plane bipolar orientations for non-separable maps.Let us give the precise definition. Given a rooted irreducible triangulation T , denote by N , E , S , W the outer vertices of T in clockwise order around the outer face, starting from the origin of the root. A transversal structure of T is an orientation and a bicoloration of the inner edges of T , say each inner edge isred or blue, such that the following conditions are satisfied, see Figure 2(b) . (T1): The edges incident to an inner vertex of T form in clockwise order: a nonempty interval ofoutgoing red edges, a nonempty interval of outgoing blue edges, a nonempty interval of ingoing rededges, and a nonempty interval of ingoing blue edges. (T2): The edges incident to N , E , S , and W are respectively ingoing red, ingoing blue, outgoing red,and outgoing blue.Transversal structures were introduced by He [ He93 ] under the name of regular edge-labellings. Theywere further investigated by the author [
Fus09 ] and have many applications in graph drawing: straight-linedrawing [
Fus09 ], visibility drawing [
KH97 ], rectangular layouts [
KH97 ]. Transversal structures characterisetriangulations of the 4-gon that are irreducible. Indeed, irreducibility is necessary, and each irreducibletriangulation admits a transversal structure [
Fus09 ]. Moreover, transversal structures are closely related toplane bipolar orientations by the following property:
Fact 2.
Let X be a transversal structure on an irreducible triangulation T having at least one inner vertex.Then the oriented map X r formed by the red edges and the vertices of T \{ W, E } is a plane bipolar poset,called the red bipolar poset of X . Similarly, the oriented map X b formed by the blue edges and the verticesof T \{ S, N } is a plane bipolar poset, called the blue bipolar poset of X . In all figures, red edges are light and blue edges are dark.
EW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 5 st (a) (b) (c) (d) Figure 3. (a) A plane bipolar orientation, (c) the associated N -avoiding plane bipolarposet, (d) the associated N -avoiding transversal structure. (a) (b) (c) Figure 4. (a) Insertion of the edges of P = φ ( O ) inside a face of O . (b) Configuration ofan inner face of P around the associated non-special vertex of O . (c) Insertion of the blueedges of X = φ ′ ( P ) inside a face of P . Proof.
The fact that X r and X b are plane bipolar orientations was proved in [ Fus09 ] (the mainargument is that a monocoloured circuit of X would have a monocolored chordal path of the other colorinside, which is not possible by Condition T1). Clearly in each inner face f of X r , the two lateral paths arenot both of length 1, as T has no multiple edge. Hence, f has at least one lateral vertex v on one side (leftor right), so f has at least one blue edge inside by Condition (T1). But, by condition (T1) again, such ablue edge goes from a left lateral vertex of f to a right lateral vertex of f . Hence both lateral paths of f have length greater than 1, so X r is a plane bipolar poset by Fact 1. (cid:3) A transversal structure is called N -avoiding if both its red and its blue bipolar posets are N -avoiding.
3. Bijections between plane bipolar orientations and N -avoiding structures This section covers the top-line of the diagram shown in Figure 1. We present a bijective mapping Φfrom plane bipolar orientations to N -avoiding transversal structures; Φ is the composition of two bijections φ and φ ′ : φ maps a plane bipolar orientation to an N -avoiding plane bipolar poset, and φ ′ completes an N -avoiding plane bipolar poset into an N -avoiding transversal structure. N -avoiding plane bipolar posets. Let O be a planebipolar orientation. We associate with O an oriented planar map P = φ ( O ) as follows, see Figure 3(a)-(c). • Add two outer edges.
Add two edges ℓ and r going from the source to the sink of O , so thatthe whole map is contained in the 2-cycle delimited on the left by ℓ and on the right by r . Theaugmented bipolar orientation is denoted e O . • Insert the vertices of P . A vertex of P , depicted in white, is inserted in the middle of each edgeof e O . The vertex inserted in the edge ℓ is denoted s P , and the vertex inserted in the edge r isdenoted t P . • Insert the edges of P . Edges of P are planarly inserted in the interior of each inner face f of e O soas to create the following adjacencies, see Figure 4(a): the vertices of P in the left lateral path of f are connected to the vertex of P in the bottomright edge of f ; and the vertices of P in the rightlateral path of f are connected to the vertex of P in the topleft edge of f . The inserted edges aredirected from the left lateral path to the right lateral path of f . ´E. Fusy Lemma 3.
Let O be a plane bipolar orientation with n edges. Then the oriented planar map P = φ ( O ) isan N -avoiding plane bipolar poset with n + 2 vertices. The source of P is the vertex s P and the sink of P isthe vertex t P .When P and the augmented bipolar orientation e O are superimposed, there is exactly one non-specialvertex v of O in each inner face f of P : the outgoing edges of v are incident to the left lateral vertices of f and the ingoing edges of v are incident to the right lateral vertices of f , see Figure 4(b). Proof.
Let us first prove that P is acyclic. Given two vertices v and v ′ of P , write v ≺ P v ′ if there isan edge of P from v to v ′ . Let e and e ′ be the edges of O that respectively correspond to v and v ′ . Notethat v ≺ P v ′ implies e ≺ ∗ e ′ . Hence the transitive closure of ≺ P corresponds to a suborder of ≤ ∗ , so P isacyclic. To prove bipolarity, observe that, for each vertex v of P distinct from s P and t P , the correspondingedge e of O has one inner face of e O on each side. Hence, by definition of φ , v has at least one ingoing edgein the face on the left of e (possibly more than one if e is the bottomright edge of that face) and at least oneoutgoing edge in the face of e O on the right of e (possibly more than one if e is the topleft edge of that face).Next, the rules for inserting the edges of P easily imply that each non-special vertex v of O gives riseto an inner face f v of P such that the edges of P outgoing at v are incident to the left lateral vertices of f v and the edges of P ingoing at v are incident to the right lateral vertices of f v , see Figure 4(b). In additionthe left lateral path of f v has length Outdeg( v ) + 1 and the right lateral path of f v has length Indeg( v ) + 1.Hence every inner face of P of the form f v has its two lateral paths of length greater than 1. Claim.
The number of non-special vertices of O is equal to the number of inner faces of P . Proof of the claim.
Write V ( M ), E ( M ) and F ( M ) for the sets of vertices, edges, and inner faces of amap M and write deg( f ) for the number of edges on the contour of an inner face f . By definition of φ , | V ( P ) | = | E ( e O ) | and E ( P ) = P f ∈ F ( e O ) (deg( f ) −
1) = 2 | E ( e O ) | − − | F ( e O ) | . By the Euler relation, | F ( P ) | = | E ( P ) | − | V ( P ) | + 1, so | F ( P ) | = | E ( e O ) | − | F ( e O ) | −
1, which is equal to ( | V ( e O ) | −
2) again by theEuler relation. Hence, the number of inner faces of P is equal to the number of non-special vertices of O . △ The claim ensures that every inner face of P is of the form f v for some non-special vertex v of O . Henceevery inner face of P has its two lateral paths of lengths greater than 1, so P is a plane bipolar poset byFact 1.Finally, the fact that P has no N -pattern is due to the two following observations: i) two edges of P with same origin are inside the same face of O , and two edges of P with same end are inside the same faceof O , hence any N -pattern must be inside a face of O ; ii) there is no N -pattern of P inside a face of O . (cid:3) As we prove next, the mapping φ is in fact a bijection and has an explicit simple inverse.Given a bipolar poset P ( N -avoiding or not), let O = ψ ( P ) be the oriented planar map defined asfollows: • Insert the vertices of O . One vertex of O is inserted in each face of P . The vertex inserted in theright special face of P is denoted s and the vertex inserted in the left special face of P is denoted t . • Insert the edges of O . Each non-special vertex v of P gives rise to an edge of O that goes from thevertex of O inside the right lateral face of v to the vertex of O inside the left lateral face of v .The correspondence between non-special vertices of O and inner faces of P , as stated in Lemma 3 andshown in Figure 4(b), ensures that ψ is exactly the procedure to recover a plane bipolar orientation from itsimage P = φ ( O ). In other words: Lemma 4.
For each plane bipolar orientation O , ψ ( φ ( O )) = O . Next we prove that ψ is also the right inverse of φ when restricted to N -avoiding structures. Lemma 5.
For any plane bipolar poset P , the oriented map O = ψ ( P ) is a plane bipolar orientation. Inaddition φ ( O ) = P if P is N -avoiding. Proof.
Let v, v ′ be two vertices of O , and let f, f ′ be the faces of P corresponding respectively to v and v ′ . Clearly, if there is an oriented path of edges of O going from v to v ′ , then f is smaller than f ′ for the left-to-right order on the faces of P . Hence, O is acyclic. In addition, each vertex v = { s, t } of O has indegree equal to the number of left lateral vertices and outdegree equal to the number of right lateralvertices of the corresponding inner face of P . Hence, s is the only source and t the only sink of O , so O isbipolar.EW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 7Now assume that P is N -avoiding, and let us now show that φ ( O ) = P . Recall that the first step of φ is to add two edges from s to t , yielding an augmented map e O . Consider the oriented map M obtained bysuperimposing e O and P , that is, there is a vertex of P in the middle of each edge of e O and a vertex of e O inside each face of P , see Figure 3(b). Call P -vertices and P -edges the vertices and edges of M that comefrom P , and call O -vertices and O -edges the vertices and edges of M that come from e O . Note that a P -edgecorresponds to an edge of P , but an O -edge corresponds to a half-edge of e O .Let v be a P -vertex. By definition of ψ (and by Property (V) of plane bipolar orientations), the edgesof M incident to v form in clockwise order: a nonempty interval of outgoing P -edges, one ingoing O -edge, anonempty interval of ingoing P -edges, and one outgoing O -edge. Hence any P -edge inside an inner face f of e O must go from a P -vertex in the left lateral path of f to a P -vertex in the right lateral path of f . Thusthe P -edges inside f are naturally ordered from down to top; in this order, two consecutive edges are saidto be meeting if they share a vertex and parallel otherwise. By a simple counting argument, the number of P -edges inside f is deg( f ) − − p ( f ), where p ( f ) is the number of parallel consecutive P -edges inside f anddef( f ) is the degree of f . As in the proof of Lemma 3, write V ( M ), E ( M ), and F ( M ) for the sets of vertices,edges, and inner faces of a map M . From the above discussion, the total number of P -edges satisfies(1) | E ( P ) | = X f ∈ F ( e O ) deg( f ) − − p ( f ) = 2 | E ( e O ) | − − | F ( e O ) | − X f ∈ F ( e O ) p ( f ) . Moreover, | E ( P ) | = | V ( P ) | + | F ( P ) | − | V ( P ) | = | E ( e O ) | by definition of φ , and | F ( P ) | = | V ( e O ) | − ψ . Hence | E ( P ) | = | E ( e O ) | + | V ( e O ) | −
3, which is equal to 2 | E ( e O ) | −| F ( e O ) | − | E ( P ) | obtained above, weconclude that P f ∈ F ( e O ) p ( f ) = 0, so all the p ( f ) are zero, so that all consecutive P -edges in any inner face of e O are meeting. As P has no N -pattern and all consecutive edges are meeting, the only possible configurationfor the P -edges inside a face of e O is the configuration shown in Figure 4(a). In other words, the P -edges arethose inserted when applying the mapping φ to O . Hence φ ( O ) = P . (cid:3) From Lemma 4 and Lemma 5 we obtain our first bijection:
Proposition 6.
For n ≥ and i ≥ , the mapping φ is a bijection between plane bipolar orientations with n edges and i non-special vertices, and N -avoiding plane bipolar posets with n non-special vertices and i innerfaces. The inverse mapping of φ is ψ . Remark.
As recalled in Section 2.2, any bipolar orientation O on a graph G = ( V, E ) (planar or not) givesrise to a partial order P = β ( O ) on E , called edge-poset, according to the precedence order of the edgesalong oriented paths (observe that no embedding is needed to define this poset). It is well known that themapping β is indeed a bijection between bipolar orientations on graphs and so called N -free posets , whichare posets with no induced N in the Hasse diagram [ HM87 ], that is, no quadruple ( v , v , v , v ) of verticessuch that the only adjacencies on these vertices are: an edge from v to v , an edge from v to v and anedge from v to v . Moreover, it is possible to enrich this construction so as to take a planar embeddinginto account: the planar embedding of the plane bipolar orientation turns to a so-called 2-realizer of theassociated poset [ dFdM96 ].Our mapping φ shares some resemblance with this classical construction, but there are significant dif-ferences. First, the bipolar poset obtained from φ might have induced N in the form of N -patterns (due tothe embedding, an induced N can appear either as an N -pattern or as an N -pattern). Second, our bijectionhas the nice feature that it operates directly on the embedding, in both directions ( φ and ψ ). The mapping φ is actually just a simple adaptation of β so as to take the embedding into account. In contrast, φ − = ψ is not just an adaptation of β − . Indeed ψ is significantly simpler, it operates on the embedding in a localway, whereas β − requires some non-local manipulations on the edge-poset. N -avoiding plane bipolar posets to N -avoiding transversal structures. The nextstep is to complete an N -avoiding plane bipolar poset into a transversal structure. The procedure, called φ ′ ,is very similar to φ . Precisely, given a plane bipolar poset P ( N -avoiding or not) the associated transversalstructure X = φ ′ ( P ) is defined as follows, see Figure 3(c)-(d): • Create the outer quadrangle.
Insert one vertex, denoted W , in the left special face of P and onevertex, denoted E , in the right special face of P . Connect W and E to the source and sink of P , ´E. Fusy denoted respectively by S and N , thus creating an outer quadrangle with vertices ( W, N, E, S ) inclockwise order. The bipolar poset augmented from P by insertion of the quadrangle —edges ofthe quadrangle oriented from S toward N — is denoted e P . • Insert the blue edges.
For each inner face f of e P , insert (in a planar way) blue edges inside f so asto create the following adjacencies, see Figure 4(c): left lateral vertices of f are connected to thebottomright lateral vertex of f , and right lateral vertices of f are connected to the topleft lateralvertex of f . The inserted blue edges inside f are directed from the left lateral vertices to the rightlateral vertices of f . In other words, f is triangulated by transversal blue edges in the unique wayavoiding blue N -patterns inside f . Lemma 7.
Let P be a plane bipolar poset. Then φ ′ ( P ) is a transversal structure whose red bipolar poset is P and whose blue bipolar poset is N -avoiding. Proof.
Clearly X := φ ′ ( P ) is a transversal structure (conditions (T1) and (T2) are satisfied) whosered bipolar poset is P . The blue bipolar poset is N -avoiding due to the two following observations: i) twoblue edges with the same origin are inside the same face of P , and two blue edges with the same end areinside the same face of P , hence any blue N -pattern must be inside a face of P ; ii) there is no blue N -patterninside a face of P . (cid:3) Call ψ ′ the mapping that associates to a transversal structure its red bipolar poset. Proposition 8.
For n ≥ and i ≥ , the mapping φ ′ is a bijection between N -avoiding plane bipolar posetswith n non-special vertices and i inner faces, and N -avoiding transversal structures with n inner verticesand n + i + 1 red edges. The inverse mapping of φ ′ is ψ ′ . Proof.
According to Lemma 7, ψ ′ ( φ ′ ( P )) = P for any plane bipolar poset P , so ψ ′ is the left inverseof φ ′ . Let us show that ψ ′ is also the right inverse of φ ′ when the mappings are restricted to N -avoidingstructures. Let X be an N -avoiding transversal structure, with X r := ψ ′ ( X ) its red bipolar poset and X b its blue bipolar poset. When applying φ ′ to X r , each inner face f of X r is triangulated by transversal blueedges going from a left lateral vertex to a right lateral vertex of f . It is easily checked that the only such N -avoiding configuration is the one where all edges are incident either to the topleft or to the bottomrightlateral vertex of f . In other words, the blue edges inside f are placed according to the insertion process of φ ′ , hence φ ′ ( P ) = X . Finally, the parameter-correspondence is due to the Euler relation, which ensures thata plane bipolar poset with n non-special vertices and i inner faces has n + i + 1 edges. (cid:3) To conclude, we have described a bijection Φ = φ ′ ◦ φ between plane bipolar orientations with n ≥ i + 2 vertices and N -avoiding transversal structures with n inner vertices and n + i + 1 red edges. Thebijection Φ operates in two steps φ and φ ′ , the intermediate combinatorial structures being the N -avoidingplane bipolar posets. Plane bipolar orientations with n edges and i non-special verticesare known to be counted by the coefficients(2) Θ n,i := 2 n ( n + 1) (cid:18) n + 1 i (cid:19)(cid:18) n + 1 i + 1 (cid:19)(cid:18) n + 1 i + 2 (cid:19) . The formula has first been proved by Rodney Baxter [
Bax01 ], guessing the answer from a recurrence satisfiedby the counting sequence; a direct computation based on the “obstinate Kernel method” is due to Bousquet-M´elou [
BM02 ]. A bijective proof has been found recently [
FPS09 ]. Summing over i the coefficients Θ n,i ,the number of plane bipolar orientations with n edges is θ n := X i n ( n + 1) (cid:18) n + 1 i (cid:19)(cid:18) n + 1 i + 1 (cid:19)(cid:18) n + 1 i + 2 (cid:19) . The numbers θ n , called the Baxter numbers (after another Baxter, Glen Baxter, whose name is given to afamily of permutations counted by θ n ), surface recurrently in the enumeration of combinatorial structures,see [ FFNO08 ] for a recent survey on “Baxter families”. Our bijections with N -avoiding structures, asdescribed in Section 3, bring new Baxter families to the surface: Proposition 9.
For n ≥ and i ≥ , the number Θ n,i counts: EW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 9 • N -avoiding plane bipolar posets with n non-special vertices and i inner faces, • N -avoiding transversal structures with n inner vertices and n + i + 1 red edges.Taking into account the parameter n only, these structures are counted by the Baxter number θ n .
4. Bijection between non-separable maps and irreducible triangulations4.1. Non-separable maps as specific plane bipolar orientations.
For rooted maps, plane bipolarorientations are always assumed to have the root edge going from the source to the sink. As alreadymentioned in Section 2.2, bipolar orientations are naturally associated with the property of non-separability;a rooted map is non-separable if and only if it admits a plane bipolar orientation. Even more is true, namely,each rooted non-separable map can be endowed with a specific plane bipolar orientation in a canonical way.Given a plane bipolar orientation, we define a left-oriented piece , shortly a LOP, as a 4-tuple ( v , v , f , f )made of two distinct vertices v , v and two distinct faces f , f of O such that the following conditions aresatisfied, see Figure 5(a): • the vertex v is the sink of f and is a left lateral vertex of f , • the vertex v is a right lateral vertex of f and is the source of f .Note that v and v must be non-special vertices and f and f must be inner faces. Fact 10 ([ dFOdMR95, OdM94 ]) . A rooted map is non-separable if and only if it admits a bipolar ori-entations. A rooted non-separable map M has a unique plane bipolar orientation with no LOP, called the minimal plane bipolar orientation of M . The terminology is due to the fact that the set of plane bipolar orientations of a fixed rooted non-separable map is a distributive lattice, the minimal element of the lattice being the bipolar orientation withno LOP, see [
OdM94 ]. The set of plane bipolar orientations with no LOP is denoted O min . Lemma 11 (link with plane bipolar posets) . Let O be a plane bipolar orientation, and let P = φ ( O ) be theassociated N -avoiding plane bipolar poset. Then O has no LOP if and only if P has no LOP. Proof.
Assume that O has a LOP ( v , v , f , f ). Let w be the vertex of P inserted in the bottomrightedge of f and w the vertex of P in the topleft edge of f . Let f ′ be the face of P corresponding to v and f ′ the face of P corresponding to v . Then ( w , w , f ′ , f ′ ) is easily checked to be a LOP of P , as shown inFigure 5(b).Conversely, assume that P has a LOP ( w , w , f ′ = f v , f ′ = f v ), with v and v the vertices of O associated respectively to the two faces f ′ and f ′ of P . Let f be the face of O containing the ingoing edgesof w , and let f be the face of O containing the outgoing edges of w . Let us first check that f and f cannot be equal. Assume f = f = f , then w is in the right lateral path and w is in the left lateral path of f ; since P = φ ( O ) and since there is a path from w to w (due to the LOP of P ), the vertices w and w must be adjacent, so the LOP of P is in fact reduced to an N -pattern, a contradiction. Thus f and f aredistinct. From the definition of ψ and the fact that w and w are not adjacent, it is easily checked that( v , v , f , f ) is a LOP of O = ψ ( P ), see Figure 5(b). (cid:3) Let P min be the set of plane bipolar posets with no LOP. Note that an N -pattern—with e its centraledge—in a plane bipolar poset induces a LOP ( v , v , f , f ) where v is the end of e , v is the origin of e , f is the face on the right of e , and f is the face on the left of e . Hence P min is also the set of N -avoidingplane bipolar posets with no LOP, which by Lemma 11 is in bijection with O min . Thus we obtain(3) O min ≃ φ P min . As we recall here from [
Fus09 ],one can endow an irreducible triangulation with a specific transversal structure in a canonical way. Givena transversal structure X , an alternating 4-cycle of X is a cycle of four edges that alternate in color, thatis, the cycle has two opposite red edges and two opposite blue edges. Given v a vertex of R , let e and e ′ bethe two edges of R incident to v , such that e ′ follows e in a counterclockwise walk around R ; e is called the left-edge of v and e is called the right-edge of v . A boundary incidence of R is the incidence of an edge e inside R with a vertex v on R ; it is called a left incidence if e has the color of the left-edge of v and a rightincidence if e has the color of the right-edge of v .0 ´E. Fusy v v f f v v f f f ′ f ′ w w w s R t R w w s R t R w (a) (b) (c) f ′ f ′ Figure 5. (a) A left-oriented piece (LOP) of a plane bipolar orientation. (b) Each LOP ina plane bipolar orientation O yields a LOP in P = φ ( O ). (c) Each right alternating 4-cyclein an N -avoiding transversal structure X yields a LOP in the red bipolar poset of X .As proved in [ Fus09 ], R is of two possible types: either all boundary incidences for R incidences areleft incidences, in which case R is called a left alternating 4-cycle , or all boundary incidences of R are rightincidences, in which case R is called a right alternating 4-cycle .Let us give a slightly different formulation. By Condition (T1), the cycle R is made of two orientedpaths of length 2 with the same origin s R and the same end t R , which are called the source and the sink of R . Let X R be X restricted to R and its interior, forgetting the colors and directions of the edges of R .Then, if R has at least one interior vertex, X R is a transversal structure. If s R is source of red edges, R is aleft alternating 4-cycle; if R is source of blue edges, R is a right alternating 4-cycle. In the degenerated casewhere R has no vertex inside—thus it has a unique edge e inside— consider the triple formed by e and thetwo edges of R of the same color as e . Then either the triple forms an N -pattern, in which case R is a leftalternating 4-cycle, or the triple forms an N -pattern, in which case R is a right alternating 4-cycle. Fact 12 ([ Fus09 ]) . A triangulation of the 4-gon is irreducible if and only if it admits a transversal structure.An irreducible triangulation admits a unique transversal structure with no right alternating 4-cycle, calledits minimal transversal structure.
Again the terminology refers to the fact that the set of transversal structures of an irreducible triangu-lation is a distributive lattice whose minimal element is the transversal structure with no right alternating4-cycle, see [
Fus09 ]. Lemma 13 (link with plane bipolar posets) . Let X be an N -avoiding transversal structure, and let X r bethe red bipolar poset of X . Then X has no right alternating 4-cycle if and only if X r has no LOP. Proof.
Assume that X has a right alternating 4-cycle R , the vertices of R in clockwise order beingdenoted ( s R , w , t R , w ), with s R the source and t R the sink of R . Let f ′ be the face of X r containing the blueedge ( w , t R ), and let f ′ be the face of X r containing the blue edge ( s R , w ). As discussed at the end of thedefinition of right alternating 4-cycles, a right alternating 4-cycle with no vertex inside yields an N -pattern.Hence, since X is N -avoiding, R has at least one vertex inside, so the faces f ′ and f ′ are distinct. It iseasily checked that ( w , w , f ′ , f ′ ) is a LOP, as shown in Figure 5(c). Conversely assume that X r has a LOP( w , w , f ′ , f ′ ). Let s R be the topleft lateral vertex of f ′ , and let t R be the bottomright lateral vertex of f ′ . Since X is N -avoiding, X is obtained from its red bipolar poset as X = φ ′ ( X r ). By definition of φ ′ (seeFigure 4(c)), in X there is a blue edge from s R to w and a blue edge from w to t R . Hence, the 4-cycle R = ( s R , w , t R , w ) is an alternating 4-cycle. As there are red edges inside R incident to w and w , s R issource of blue edges, so R is a right alternating 4-cycle. (cid:3) Let T min be the set of transversal structures with no right alternating 4-cycle (minimal transversalstructures). Note that an N -pattern in a transversal structure yields a right alternating 4-cycle: the cycledelimited by the faces on both sides of the central edge of the N -pattern. Hence T min is also the set ofEW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 11 N -avoiding transversal structures with no right alternating 4-cycle, which by Lemma 13 is in bijection with P min . Thus we obtain:(4) P min ≃ φ ′ T min . To conclude, we have O min ≃ φ P min ≃ φ ′ T min . Hence Φ = φ ′ ◦ φ specializes into a bijection between minimal plane bipolar orientations and minimaltransversal structures. Since minimal plane bipolar orientations identify to (rooted) non-separable maps andminimal transversal structures identify to (rooted ) irreducible triangulations, we obtain a bijection, called F , between these two map families. Precisely, given a rooted non-separable map M with at least 2 edges T := F ( M ) is obtained as follows: • endow M with its minimal plane bipolar orientation O , the root edge being deleted, • compute the transversal structure X associated to O : X := Φ( O ), • return T as the irreducible triangulation underlying X , rooted at the edge going from N to W ,and M = F − ( T ) is obtained as follows: • endow T with its minimal transversal structure, • compute the plane bipolar orientation O associated to X : O := ψ ( ψ ′ ( X )), • return the rooted non-separable map M underlying O ( M receives an additional root edge goingfrom the source to the sink). Theorem 14.
For n ≥ , the mapping F is a bijection between rooted non-separable maps with n edgesand irreducible triangulations with n + 3 vertices. Proof.
The bijection results from O min ≃ T min , as discussed above. The parameter-correspondence isinherited from the parameter-correspondences of φ and φ ′ . (cid:3) Brown and Tutte [
BT64 ] used the recursive approach intro-duced by Tutte to show that the number of rooted non-separable maps with n + 1 edges and i + 2 verticesis(5) Λ n,i = ( n + i )!(2 n − i − i + 1)!( n − i )!(2 i + 1)!(2 n − i − . And the number of rooted non-separable with n + 1 edges is λ n = P n − i =0 Λ n,i , which simplifies to(6) λ n = 2(3 n )!( n + 1)!(2 n + 1)! . Bijective proofs have been given later on, the first one being a correspondence with ternary trees bearinglabels so as to satisfy a positivity condition [
JS98 ]. A more direct construction has been described bySchaeffer in his PhD [
Sch98 ], as a 4-to-(2 n + 2) correspondence between ternary trees with n nodes androoted non-separable maps with n + 1 edges. The correspondence is based on local “closure” operations ona ternary tree, with the effect of closing one by one the faces of the associated map.Our bijections relating maps via specific combinatorial structures, as described in Section 4, yields thefollowing counting results: Proposition 15.
For n ≥ and i ≥ , the number Λ n,i counts: • rooted non-separable maps with n + 2 vertices, i + 2 faces, and whose minimal plane bipolar orien-tation (forgetting the root-edge) is a plane bipolar poset, • rooted irreducible triangulations with n inner vertices and whose minimal transversal structure has n + i + 1 red edges.Taking into account the parameter n only, these map families are counted by λ n . Remark.
As for rooted non-separable maps, there exists a simple 4-to-(2 n + 2) correspondence betweenternary trees with n nodes and rooted irreducible triangulations with n inner vertices [ Fus09 ], again basedon “closure” operations on the tree.It turns out that our bijection F puts in correspondence a non-separable map M and an irreducibletriangulation T that have the same underlying (unrooted) ternary tree. This is no surprise, as the ternary2 ´E. Fusy tree associated to M ( T ) is a spanning tree that arises from deleting specific edges in the minimal bipolarorientation of M (minimal transversal structure of T , resp.), whereas F results from φ matching minimalplane bipolar orientations with minimal transversal structures. Hence, F can be considered as a directbijective construction on maps that has the same effect as computing the ternary tree associated to a non-separable map and then closing the tree into an irreducible triangulation.
5. Bijection between loopless maps and triangulations5.1. Decomposing loopless maps into non-separable components.
It is well known in graphtheory that a connected graph G is decomposed into a collection of non-separable components called the blocks , which are the maximal non-separable subgraphs of G [ HP73, MT01 ]. Conversely, the process ofgluing non-separable graphs at common vertices in a tree-like fashion yields any connected graph in a uniqueway. This classical decomposition readily adapts to rooted loopless maps [
Tut63, GJ83 ]. For our purposeit proves convenient to define the size | M | of a rooted loopless map as its number of edges including the root.In this section we consider the vertex-map—made of a unique vertex and no edge—as a rooted loopless map.In contrast, non-separable maps are required to have at least one edge. Fact 16.
The following process: • take a rooted non-separable map C , and order the corners of C in a canonical way as α , . . . , α | C | ; • take a collection M , . . . , M | C | of rooted loopless maps; for each ≤ i ≤ | C | , insert M i in thecorner α i of C , merging the outer face of M i with the face incident to α i and the root vertex of M i with the vertex incident to α i ,yields a rooted loopless map, and each rooted loopless map M is obtained in a unique way from this process.The map C is called the core-map of M , and the ( | C | +1 )-tuple D ( M ) = ( C ; M , . . . , M | C | ) is called the decomposition-tuple of M . (Note that the size of M is the sum of the sizes of the maps in the decomposition-tuple.) Similarly as for loopless maps,there exists a classical decomposition of triangulations into components of higher connectivity: each trian-gulation on the topological sphere is obtained from a collection of 4-connected triangulations (which aretriangulations where all 3-cycles are facial) glued at common triangles in a tree-like fashion. The same idea—decomposition at separating 3-cycles— works as well to decompose rooted triangulations of the 4-gon intorooted irreducible components [
Tut62 ]. First let us introduce some terminology. Given a rooted triangula-tion T of the 4-gon, denote by N, E, S, W the outer vertices of T in clockwise order around the outer face,with N the root-vertex; then T is called WE -diagonal if W and E are adjacent, T is called SN -diagonal if S and N are adjacent, and T is called non-diagonal otherwise. Notice that the two diagonal cases aredisjoint by planarity of T . The unique rooted irreducible triangulation that is WE -diagonal is the quadrangle( N, E, S, W ) split by a diagonal (
W, E ); this map is called the WE -link-map . Similarly the unique rootedirreducible triangulation that is SN -diagonal is the quadrangle ( N, E, S, W ) split by a diagonal (
S, N ); thismap is called the SN -link-map . For our purpose it is convenient to define the size of a triangulation or of atriangulation of the 4-gon T as || T || := ( | V ( T ) | − || T || is the number of inner vertices if T is a triangulation and 2 || T || is the number of inner faces if T is a triangulation of the 4-gon, by the Eulerrelation. Fact 17.
The following process: • take a rooted irreducible triangulation I , and order the inner faces of I in a canonical way as f , . . . , f || I || ; • take a collection T , . . . , T || I || of rooted triangulations; for each ≤ i ≤ || I || , substitute f i by T i in a canonical way (e.g., the outer triangle of T i fits with the contour of f i and the root vertex of T i fits with a distinguished vertex of f i )yields a rooted triangulation of the 4-gon; and each rooted triangulation T of the 4-gon is obtained in a uniqueway from this process.The map I is called the core-triangulation of T ; T is non-diagonal if I has at least one inner vertex, is WE -diagonal if I is the WE -link-map, and is SN -diagonal if I is the SN -link-map. The (2 || I || + 1) -tuple D ( T ) = ( I ; T , . . . , T || I || ) is called the decomposition-tuple of T . (Note that the size of T is the sum of thesizes of the maps in the decomposition-tuple.) EW BIJECTIVE LINKS ON PLANAR MAPS VIA ORIENTATIONS 13As a corollary we can describe the decomposition-tuples corresponding to rooted triangulations. Indeed,in each size k ≥
1, there is a simple bijection between rooted triangulations and rooted triangulations of the4-gon that are not
W E -diagonal. Starting from a rooted triangulation T , delete the edge following the rootin counterclockwise order around the outer face; the obtained map is a triangulation of the 4-gon, and W and E are not adjacent, since the unique edge connecting them in T has been deleted. Conversely, startingfrom a rooted triangulation e T of the 4-gon that is not W E -diagonal, add an edge between E and W . Since W and E are not adjacent in e T , the obtained map T has no multiple edges (nor loops) and has clearly allfaces of degree 3, so T is a triangulation. Corollary 18.
In each size k ≥ , rooted triangulations can be identified with rooted triangulations of the4-gon that are not W E -diagonal, which themselves identify to decomposition-tuples whose core-triangulationis not the
W E -link-map.
Our size-preserving bijection, called F , between rooted loopless maps and rootedtriangulations is defined recursively from the bijection F . Recall that F maps a non-separable map of sizeat least 2 (at least one non-root edge) to an irreducible triangulation of the same size (at least one innervertex). It is convenient here to augment the correspondence F into a bijection e F with one object of size 1on each side: the edge-map—made of two vertices connected by an edge—corresponds to the SN -link-map.We denote by L the family of (rooted) loopless maps including the vertex-map, by L ≤ n the set of looplessmaps of size at most n , and by L n the set of loopless maps of size n . Similarly, denote by T the family of(rooted) triangulations including the triangle, by T ≤ n the set of triangulations of size at most n , and by T n the set of triangulations of size n . Finally, denote by N the family of (rooted) non-separable maps with atleast one edge and by I the family of (rooted) irreducible triangulations with at least one inner vertex. Notethat e F is a size-preserving bijection between N and I ∪ { SN -link-map } .The size-preserving bijection F between L and T is specified recursively as follows. First, the vertex-map—the unique rooted loopless map M such that | M | = 0—is mapped by F to the triangle-map—theunique rooted triangulation T such that || T || = 0. Given n ≥
0, assume that F is a well defined bijectionfrom L ≤ n to T ≤ n , that is size-preserving: || F ( M ) || = | M | for every M ∈ L ≤ n .Let us now extend the bijection F to maps of size n + 1. By Fact 16, a tuple ( C ; M , . . . , M | C | )corresponds to a loopless map in L n +1 if and only if C is in N , the M i ’s are in L , and | C | + P i | M i | = n + 1,in which case all the M i ’s must be in L ≤ n . Let DL n +1 be the set of such decomposition-tuples, notethat DL n +1 ≃ L n +1 . Similarly, by Fact 17 and Corollary 18, a tuple ( I ; T , . . . , T || I || ) corresponds to atriangulation in T n +1 if and only if I is in I ∪ { SN − link − map } , the T i ’s are in T , and || I || + P i || T i || = n + 1, in which case all the T i ’s must be in T ≤ n . Let DT n +1 be the set of such decomposition-tuples,note that DT n +1 ≃ T n +1 . Since e F is a size-preserving bijection between N and I ∪ { SN -link-map } and since, by induction, F is a bijection between L ≤ n and T ≤ n , we obtain a bijection DL n +1 ≃ DT n +1 :for ( C ; M , . . . , M | C | ) ∈ DL n +1 , the associated tuple in DT n +1 is ( e F ( C ); F ( M ) , . . . , F ( M | C | )). Since DL n +1 ≃ L n +1 and DT n +1 ≃ T n +1 , we conclude that L n +1 is in bijection with T n +1 , and we call F thebijection extended to L ≤ n +1 . Theorem 19.
For n ≥ , the mapping F is a bijection between rooted loopless maps with n edges and rootedtriangulations with n inner vertices. Remark.
Let us make a few comments on this bijection. Denote by L n,k the family of rooted looplessmaps with n edges and root-vertex of degree k , and denote by T n,k the family of rooted triangulations with n inner vertices and root-vertex of degree k + 2. The equality |L n | = |T n | is known for long [ Tut62, Tut73 ],as both |L n | and |T n | are equal to(7) a n := 2(4 n + 1)!( n + 1)!(3 n + 2)! . A first bijective proof of |L n | = |T n | , more precisely of |L n,k | = |T n,k | , has been found by Wormald [ Wor80 ],based on the observation that the generating trees (induced by root-edge-deletion) of L = ∪ n,k L n,k and of T = ∪ n,k T n,k are isomorphic. The bijection is thus recursive; a rooted loopless map is first completely decom-posed to find out the place it occupies in the generating tree of L , then the associated rooted triangulationis the one occupying the same place in the generating tree of T .4 ´E. Fusy Our bijection has the original feature of being a mixing of a recursive construction (the parallel recursivedecompositions into non-separable/irreducible components) and a direct construction (the mapping F tomatch non-separable/irreducible components). The advantage is that, in practice, there should be fewerlevels of recursion in our bijection than in Wormald’s one. Indeed, as shown in [ BFSS01 ], a typical looplessmap is made of a giant non-separable component —of linear size— having small connected componentsattached in each corner.Let us also mention that direct bijective proofs of |T n | = a n have been found recently [ PS06, FPS08 ],again based on closure operations on trees. However, no bijection with trees is known yet to explain |L n | = a n .It would be interesting to find out such a construction for loopless maps, and hopefully derive from it a direct(nonrecursive) bijection between L n and T n . Acknowledgements.
I am very grateful to Gilles Schaeffer for his encouragements to find a new bijectionbetween loopless maps and triangulations. I thank the anonymous referee for a very detailed report thathas led to a significant improvement of the presentation of the results. This work has also benefited frominteresting discussions with Mireille Bousquet-M´elou, Nicolas Bonichon, and Nick Wormald.
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