BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES OLIVIER BERNARDI AND ÉRIC FUSY Abstract. We present bijections for planar maps with bounaries. In particular, we obtain bijections for triangulations an quarangulations of the sphere with bounaries of prescribe lengths. For triangulations we recover the beautiful factorize formula obtaine by Krikun using a (technically involve) generating function approach. The analogous formula for quarangulations is new. We also obtain a far-reaching generalization for other face-egrees. In fact, all the known enumerative formulas for maps with bounaries are prove bijectively in the present article (an several new formulas are obtaine). Our metho is to show that maps with bounaries can be enowe with certain canonical orientations, making them amenable to the master bijection approach we evelope in previous articles. As an application of our enumerative formulas, we note that they provie an exact solution of the imer moel on roote triangulations an quarangulations.. Introuction In this article, we present bijections for planar maps with bounaries. Recall that a planar map is a ecomposition of the 2-imensional sphere into vertices, eges, an faces, consiere up to continuous eformation (see precise efinitions in Section 2). We eal exclusively with planar maps in this article an call them simply maps from now on. A map with bounaries is a map with a set of istinguishe faces calle bounary faces which are pairwise vertex-isjoint, an have simple-cycle contours (no pinch points). We call bounaries the contours of the bounary faces. We can think of the bounary faces as holes in the sphere, an maps with bounaries as a ecomposition of a sphere with holes into vertices, eges an faces. A triangulation with bounaries (resp. quarangulation with bounaries) is a map with bounaries such that every non-bounary face has egree 3 (resp. 4). The main results obtaine in this article are bijections for triangulations an quarangulations with bounaries. The bijection establishes a corresponence between these maps an certain types of plane trees. This, in turns, easily yiels factorize enumeration formulas with control on the number an lengths of the bounaries. In the case of triangulations, the enumerative formula ha been establishe by Krikun [0] (by a technically involve guessing/checking generating function approach). The case of quarangulations is new. We also present a far-reaching generalization for maps with other face-egrees. The strategy we apply is to aapt to maps with bounaries the master bijection approach we evelope in [2, 3] for maps without bounaries. Roughly speaking, this strategy reuces the problem of fining bijections, to the problem of exhibiting canonical orientations characterizing these classes of maps. Let us now state the enumerative formulas erive from our bijections for triangulations an quarangulations. We call a map with bounary multi-roote if the r bounary faces are labele with istinct numbers in [r] = {,..., r}, an each one has a marke corner; see Figure. For m 0 an a,..., a r positive integers, we enote T (m; a,..., a r ) (resp. Q(m; a,..., a r )) the set of multi-roote triangulations (resp. Date: February 0, 207. Department of Mathematics, Braneis University, Waltham MA, USA, bernari@braneis.eu. LIX, École Polytechnique, Palaiseau, France, fusy@lix.polytechnique.fr.
2 O. BERNARDI AND É. FUSY 2 2 3 3 Figure. Left: a quarangulation in Q[3; 4, 2, 6]. Right: a triangulation in T [3; 2,, 3]. quarangulations) with r bounary faces, an m internal vertices (vertices not on the bounaries), such that the bounary labele i has length a i for all i [r]. In 2007 Krikun prove the following result: Theorem. (Krikun [0]). For m 0 an a,..., a r positive integers, () T [m; a,..., a r ] = 4k (e 2)!! r ( ) 2ai a i, m!(2b + k)!! a i where b := r i= a i is the total bounary length, k := r + m 2, an e = 2b + 3k is the number of eges (an the notation n!! stans for (n )/2 i=0 (n 2i)). We obtain a bijective proof of this result, an also prove the following analogue: Theorem.2. For m 0 an a,..., a r positive integers, (2) Q[m; 2a,..., 2a r ] = 3k (e )! m!(3b + k)! i= r i= ( ) 3ai 2a i, a i where b := r i= a i is the half-total bounary length, k := r + m 2, an e = 3b + 2k is the number of eges. Equations () an (2) are generalizations of classical formulas. Inee, the oubly egenerate case m = 0 an r = of () gives the well-known Catalan formula for the number of triangulations of a polygon without interior points T [0; a] = Cat(a 2) = (2a 4)! (a )!(a 2)!. Similarly, the case m = 0 an r = of (2) gives the 2-Catalan formula for the number of quarangulations of a polygon without interior points Q[0; 2a] = (3a 3)! (a )!(2a )!. The case r =, a = of (2) is alreay non-trivial as it gives the well-known formula for the number of roote quarangulations with m + 2 vertices (upon seeing the root-ege as blown into a bounary face of egree 2): Q[m; 2] = 2 3m (2m)! m!(m + 2)!. More generally, the case r = of (2) yiels Q[m; 2a] = 3m (3a + 2m 3)! m!(3a + m )! (3a)! (2a )!a!, which is the formula given in [6, Eq.(2.2)] for the number of roote quarangulations with one simple bounary of length 2a, an m internal vertices. Similarly, the case r = of () yiels the formula for the
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 3 number of roote triangulations with one simple bounary of length a, but it seems that this formula was not known prior to [0]. Lastly, in Section 5 we use the special case of () an (2) where all the bounaries have length 2 in orer to solve the imer moel on triangulations an quarangulations. As a sie remark, let us iscuss the counterparts of (.) an (2) when we remove the conition for the bounaries to be simple an pairwise isjoint. Let T [n; a,, a r ] (resp. Q[n; a,..., a r ]) be the set of maps with n + r faces, n faces of egree 3 (resp. 4) an r istinguishe faces labele,..., r of respective egrees a,..., a r, each having a marke corner. It is easy to euce from Tutte s slicings formula [7] that Q[n; 2a,..., 2a r ] = (e )! v!n! 3 n r i= ( ) 2ai 2a, a i where v = n + 2 + r i= (a i ) is the total number of vertices, an e = 2n + r i= a i is the total number of eges. However no factorize formula shoul exist for T [n; a,..., a r ], since the formula for r = is alreay complicate []. As mentione above, we have also generalize our results to other face egrees. For these extensions, there is actually a necessary girth conition to take into account in orer to obtain bijections. Precisely, we efine a notion of internal girth for plane maps with bounaries. The internal girth coincies with the girth when the map has at most one bounary (but can be larger than the girth in general). For any integer, we obtain a bijection for maps with bounaries having internal girth, an non-bounary faces of egrees in {, +, + 2} (with control on the number of faces of each egree). For =, the internal girth conition is voi, an restricting the non-bounary faces to have egree + 2 = 3 gives our result for triangulations with bounaries. For = 2, the internal girth conition is voi for bipartite maps, an restricting the non-bounary faces to have egree + 2 = 4 gives our result for bipartite quarangulations with bounaries. For the values of 3, the case of a single bounary with all the internal face of egree correspons to the results obtaine in [2] (bijections for -angulations of girth 3 with at most one bounary). For = 2, the case of a single bounary with all the internal face of egree 3 gives a bijection for loopless triangulations (i.e. triangulations of girth at least 2) with a single bounary an we recover the counting formula of Mullin [2]. Hence, our bijections cover the cases of triangulations with a single bounary with girth at least, for {, 2, 3} (for girth we give the first bijective proof, while for girth 2 the first bijective proof was given in [3] an for girth 3 it was given in [4], an generalize to -angulations in []). Furthermore, in Theorem 6.2 we give generalizations of these results in the form of multivariate factorize counting formulas, analogous to Krikun formula (), for the classes of triangulations of internal girth = 2 an = 3. Lastly, we give multivariate factorize counting formulas for the classes of quarangulations of internal girth = 4 thereby generalizing the formula of Brown [7] for simple quarangulations with a single bounary. In fact, all the known counting formulas for maps with bounaries are prove bijectively in the present article. This article is organize as follows. In Section 2 we set our efinitions about maps, an aapt the master bijection establishe in [2] to maps with bounaries. In Section 3, we efine canonical orientations for quarangulations with bounaries, an obtain a bijection with a class of trees calle mobiles (the case where at least one bounary has size 2 is simpler, while the general case requires to first cut the map into two pieces). In Section 4 we treat similarly the case of triangulations. In Section 5, we count mobiles an obtain () an (2). We also erive from our formulas (both for coefficients an generating functions) exact solutions of the imer moel on roote quarangulations an triangulations. In Section 6 we unify an exten the results (orientations, bijections, an enumeration) to more general face-egree conitions. In Section 7, we prove the existence an uniqueness of the neee canonical orientations for maps with bounaries. Lastly, in Section 8, we iscuss aitional results an perspectives.
4 O. BERNARDI AND É. FUSY 2. Maps an the master bijection In this section we set our efinitions about maps an orientations. We then recall the master bijection for maps establishe in [2], an aapt it to maps with bounaries. 2.. Maps an weighte biorientations. A map is a ecomposition of the 2-imensional sphere into vertices (points), eges (homeomorphic to open segments), an faces (homeomorphic to open isks), consiere up to continuous eformation. A map can equivalently be efine as a rawing (without ege crossings) of a connecte graph in the sphere, consiere up to continuous eformation. Each ege of a map is thought as mae of two half-eges that meet in its mile. A corner is the region between two consecutive half-eges aroun a vertex. The egree of a vertex or face x, enote eg(x), is the number of incient corners. A roote map is a map with a marke corner c 0 ; the incient vertex v 0 is calle the root vertex, an the half-ege (resp. ege) following c 0 in clockwise orer aroun v 0 is calle the root half-ege (resp. root ege). A map is sai to be bipartite if the unerlying graph is bipartite, which happens precisely when every face has even egree. A plane map is a map with a face istinguishe as its outer face. We think about plane maps, as rawn in the plane, with the outer face being the infinite face. The non-outer faces are calle inner faces; vertices an eges are calle outer or inner epening on whether they are incient to the outer face or not; an half-ege is inner if it belongs to an inner ege an outer if it belongs to an outer ege. The egree of the outer face is calle the outer egree. A biorientation of a map M is the assignment of a irection to each half-ege of M, that is, each half-ege is either outgoing or ingoing at its incient vertex. For i {0,, 2}, an ege is calle i-way if it has i ingoing half-eges. An orientation is a biorientation such that every ege is -way. If M is a plane map enowe with a biorientation, then a ccw cycle (resp. cw-cycle) of M is a simple cycle C of eges of M such that each ege of C is either 2-way or -way with the interior of C on its left (resp. on its right). The biorientation is calle minimal if there is no ccw cycle, an almost-minimal if the only ccw cycle is the outer face contour (in which case the outer face contour must be a simple cycle). For u, v two vertices of M, v is sai to be accessible from u if there is a path P = u 0, u,..., u k of vertices of M such that u 0 = u, u k = v, an for i [..k ], the ege (u i, u i+ ) is either -way from u i to u i+ or 2-way. The biorientation is sai to be accessible from u if every vertex of M is accessible from u. A weighte biorientation of M is a biorientation of M where each half-ege is assigne a weight (in some aitive group). A Z-biorientation is a weighte biorientation such that weights at ingoing half-eges are positive integers, while weights at outgoing half-eges are non-positive integers. 2.2. Master bijection for Z-bioriente maps. We first efine the families of bioriente maps involve in the master bijection. Let be a positive integer. We efine O as the set of plane maps of outer egree enowe with a Z-biorientation which is minimal an accessible from every outer vertex, an such that every outer ege is either 2-way or is -way with an inner face on its right. We efine O as the set of plane maps of outer egree enowe with a Z-biorientation which is almost-minimal an accessible from every outer vertex, an such that outer eges are -way with weights (0, ), an each inner half-ege incient to an outer vertex is outgoing. Next, we efine the families of trees involve in the master bijection. We call mobile an unroote plane tree with two kins of vertices, black vertices an white vertices (vertices of the same color can be ajacent), where each corner at a black vertex possibly carries aitional angling half-eges calle bus; see Figure 3 (right) for an example. The excess of a mobile is efine as the number of half-eges incient to a white vertex, minus the number of bus. A weighte mobile is a mobile where each half-ege, except for bus, is assigne a weight. A Z-mobile is a weighte mobile such that weights of half-eges incient to white vertices are positive integers, while weights at half-eges incient to black vertices are non-positive integers. For Z, we enote by B the set of Z-mobiles of excess.
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 5 w w w w w w w w w w w w Figure 2. The local rule performe at each ege (0-way, -way or 2-way) in the master bijection Φ. 0-2 -3-5 2-0 2 0 0 4-3 3 2 0-0 -2 2 0-5 - 2-3 0-3 4 2-3 - 3 0 2-5 0 4-2 -4 3-3 - 2 0 3 2 3 0 3 - -5 2-4 0-3 -4-2 2 0 2 3 Figure 3. The master bijection from a Z-bioriente plane map in O to a Z-mobile of excess (the top example has = 4, the bottom-example has = 5). Let Z\{0}. We now recall the master bijection Φ introuce in [2] between O an B. For O O, we obtain a mobile T B by the following steps (see Figure 3 for examples):
6 O. BERNARDI AND É. FUSY () insert a black vertex in each face (incluing the outer face) of O; (2) apply the local rule of Figure 2 (which involves a transfer of weights) to each ege of O; (3) erase the original eges of O an the black vertex b inserte in the outer face of O; if > 0 erase also the bus at b, if < 0 erase also the outer vertices of O an the eges from b to each of the outer vertices. Theorem 2. ([2]). For Z\{0}, the mapping Φ is a bijection between O an B. The master bijection has the nice property that several parameters of a Z-bioriente map M O can be rea on the associate Z-mobile T = Φ(M). We efine the weight (resp. the inegree) of a vertex v M as the total weight (resp. total number) of ingoing half-eges at v, an we efine the weight of a face f M as the total weight of the outgoing half-eges having f on their right. For a vertex v T, we efine the egree of v as the number of half-eges incient to v (incluing bus if v is black), an we efine the weight of v as the total weight of the half-eges (excluing bus) incient to v. It is easy to see that if M O an T = Φ(M), then every inner face of M correspons to a black vertex in T of same egree an same weight, for > 0 (resp. < 0), every vertex (resp. every inner vertex) v M correspons to a white vertex v T of the same weight an such that the inegree of v equals the egree of v. 2.3. Aaptation of the master bijection to maps with bounaries. A face f of a map is sai to be simple if the number of vertices incient to f is equal to the egree of f (in other wors there is no pair of corners of f incient to the same vertex). A map with bounaries is a map M where the set of faces is partitione into two subsets: bounary faces an internal faces, with the constraint that the bounary faces are simple, an the contours of any two bounary faces are vertex-isjoint; these contour-cycles are calle the bounaries of M. Eges (an similarly half-eges an vertices) are calle bounary eges or internal eges epening on whether they are on a bounary or not. If M is a plane map with bounaries, whose outer face is a bounary face, then the contour of the outer face is calle the outer bounary an the contours of the other bounary faces are calle inner bounaries. For M a map with bounaries, a Z-biorientation of M is calle consistent if the bounary eges are all -way with weights (0, ) an have the incient bounary face on their right. For Z\{0}, we enote by Ô the set of plane maps with bounaries enowe with a consistent Z-biorientation, such that the outer face is a bounary face for < 0 an an internal face for > 0, an when forgetting which faces are bounary faces, the unerlying Z-bioriente plane map is in O. A bounary mobile is a mobile where every corner at a white corner might carry aitional angling halfeges calle legs. White vertices having at least one leg are calle bounary vertices. The egree of a white vertex v is the number of non-leg half-eges incient to v. The excess of a bounary mobile is efine as the number of half-eges incient to a white vertex (incluing the legs) minus the number of bus. A bounary Z-mobile is a bounary mobile where the half-eges ifferent from bus an legs carry weights in Z such that half-eges at white vertices have positive weights while half-eges at black vertices have non-positive weights. For Z, we enote by B the set of bounary Z-mobiles of excess. We can now specialize the master bijection. For O Ô, let T = Φ(O) be the associate Z-mobile. Note that each inner bounary face f of O of egree k yiels a black vertex v of egree k in T such that v has no bu, an the k neighbors w,..., w k of v are the white vertices corresponing to the vertices aroun f. We perform the following operation represente in Figure 4: we insert one leg at each corner of v, then contract the eges incient to v, an finally recolor b as white. Doing this for each inner bounary we obtain (without loss of information) a bounary Z-mobile T of the same excess as T, calle the reuction of T. We enote by Φ the mapping such that Φ(O) = T.
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 7...... v......... =............... (a) (b) Figure 4. Reuction operation at the black vertex v corresponing to an inner bounary face in O Ô. We now argue that Φ is a bijection between Ô an B. For a bounary mobile T, the expansion of T is the mobile T obtaine from T by applying to every bounary vertex the process of Figure 4 in reverse irection: a bounary vertex with k legs yiels in T a istinguishe black vertex of egree k with no bus, an with only white neighbors. Note that, if T has non-zero excess an if O O enotes the Z-bioriente plane map associate to T by the master bijection, then each istinguishe face f O (i.e., a face associate to a istinguishe black vertex of T ) is simple; inee if k enotes the egree of f, the corresponing black vertex v T has k white neighbors, which thus correspon to k istinct vertices incient to f. In aition the contours of the istinguishe inner faces are isjoint since the expansions of any two istinct bounary vertices of T are vertex-isjoint in T. Lastly, for Z, the outer face is simple an isjoint from the contours of the inner istinguishe faces (inee the vertices aroun an inner istinguishe face of O are all present in T, hence are inner vertices of O). We thus conclue that O belongs to Ô, upon seeing the istinguishe faces (incluing the outer face for Z ) as bounary faces. The following statement summarizes the previous iscussion: Theorem 2.2. The master bijection Φ aapte to consistent Z-biorientations is a bijection between Ô an B for each Z\{0}. The bijection Φ is illustrate in Figure 5. As before, several parameters can be tracke through the bijection. For a map M with bounaries enowe with a consistent Z-biorientation, we efine the weight (resp. the inegree) of a bounary C as the total weight (resp. total number) of ingoing half-eges incient to a vertex of C but not lying on an ege of C. For a bounary Z-mobile, we efine the weight of a white vertex v as the total weight of the half-eges (excluing legs) incient to v. It is easy to see that if O Ô an T = Φ(O), then every internal inner face of O correspons to a black vertex in T of same egree an same weight, every internal vertex v O correspons to a non-bounary white vertex v T of the same weight an such that the inegree of v equals the egree of v, every inner bounary of length k, inegree r, an weight j in O correspons to a bounary vertex in T with k legs, egree r, an weight j.
8 O. BERNARDI AND É. FUSY - -2 2-3 -4-2 3 4 - -5 0-2 -2-3 2-3 - - -5-2 2 0-4 4 2-4 3 - -2-3 -5 3 0-4 -2 4 3-2 - 4 3-2 -5 - -2-3 -4-3 0 4 3 Figure 5. The master bijection Φ applie to two Z-bioriente plane maps in Ô, with = 4 for the top-example, an = 5 for the bottom-example. The weights of bounaryeges, which are always (0, ) by efinition, are not inicate. 3. Bijections for quarangulations with bounaries In this section we obtain bijections for quarangulations with bounaries, that is, maps with bounaries such that every internal face has egree 4. We start with the simpler case where one of the bounaries has egree 2 before treating the general case. 3.. Quarangulations with at least one bounary of length 2. We enote by D the class of bipartite quarangulations with bounaries, with a marke bounary face of egree 2. We think of maps in D as plane maps by taking the marke bounary as the outer face. For M D, we call -orientation of M a consistent Z-biorientation with weights in {, 0, } such that: every internal ege has weight 0 (hence is either 0-way with weights (0, 0) or -way with weights (, )), every internal face (of egree 4) has weight, every internal vertex has weight (an inegree), every inner bounary of length 2i has weight (an inegree) i +, an the outer bounary (of length 2) has weight (an inegree) 0.
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 9 Proposition 3.. Every map M D has a unique -orientation in Ô 2. We call it its canonical biorientation. The proof of Proposition 3. is elaye to Section 7. We enote by T the set of bounary mobiles associate to maps in D (enowe with their canonical biorientation) via the master bijection for maps with bounaries. By Theorem 2.2, these are the bounary mobiles with weights in {, 0, } satisfying the following properties: every ege has weight 0 (hence, is either black-black of weights (0, 0), or black-white of weights (, )), every black vertex has egree 4 an weight (hence has a unique white neighbor), for all i 0, every white vertex of egree i + carries 2i legs. We omit the conition that the excess is 2, because it can easily be checke to be a consequence of the above properties. (a) (b) (c) Figure 6. (a) A map in D enowe with its canonical biorientation (the -way eges are inicate as irecte eges, the 0-way eges are inicate as unirecte eges, an the weights, which are uniquely inuce by the biorientation, are not inicate). (b) The quarangulation superimpose with the corresponing mobile. (c) The reuce bounary mobile (with 2i legs at each white vertex of egree i + ), where again the weights (which are uniquely inuce by the mobile) are not inicate. To summarize, Theorem 2.2 an Proposition 3. yiel the following bijection (illustrate in Figure 6) for bipartite quarangulations with a istinguishe bounary of length 2. Theorem 3.2. The set D of quarangulations with bounaries is in bijection with the set T of Z-mobiles via the master bijection Φ. If M D an T T are associate by the bijection, then each inner bounary of length 2i in M correspons to a white vertex in T of weight (an egree) i +, an each internal vertex of M correspons to a white leaf in T.
0 O. BERNARDI AND É. FUSY 3.2. Quarangulations with arbitrary bounary lengths. For a, we enote by D (2a) the set of bipartite quarangulations with bounaries with a marke bounary face of egree 2a. In the previous section we obtaine a bijection for D (2) = D. In orer to get a bijection for D (2a) when a >, we will nee to first mark an ege an ecompose our marke maps into two pieces before applying the master bijection to each piece. Let D (2a) be the set of maps obtaine from maps in D (2a) by also marking an ege (either an internal ege or a bounary ege). Let A (2a) be the set of bipartite maps with a marke bounary face of egree 2a an a marke internal face of egree 2, such that all the non-marke internal faces have egree 4. We also enote by A (2a) the set of maps obtaine from maps in A (2a) by marking a corner in the marke bounary face. Given a map M in D (2a), we obtain a map M in A (2a) by opening the marke ege into an internal face of egree 2. This operation, which we call ege-opening is clearly a bijection for a > : Lemma 3.3. For all a >, the ege-opening is a bijection between D (2a) number of internal vertices an the bounary lengths. an A (2a) which preserves the Note however that A (2) contains a map ɛ with 2 eges (a 2-cycle separating a bounary an an internal face) which is not obtaine from a map in D (2), so that the bijection is between D (2) an A (2) \ {ɛ}. We will now escribe a canonical ecomposition of maps in A (2a) illustrate in Figure 7(a)-(b). Let M be in A (2a), an let f s be the marke bounary face. Let C be a simple cycle of M, an let R C an L C be the regions boune by C containing f s an not containing f s respectively. The cycle C is sai to be blocking if C has length 2, the marke internal face is in L C, an any bounary face incient to a vertex of C is in R C. Note that the contour of the marke internal face is a blocking cycle. It is easy to see that there exists a unique blocking cycle C such that L C is maximal (that is, contains L C for any blocking cycle C ). We call C the maximal blocking cycle of M. The maximal blocking cycle is inicate in Figure 7(a). The map M is calle reuce if its maximal blocking cycle is the contour of the marke internal face, an we enote by B (2a) an B (2a) the subsets of A (2a) an A (2a) corresponing to reuce maps. We now consier the two maps obtaine from a map M in A (2a) by cutting the sphere along the maximal blocking cycle C, as illustrate in Figure 7(b). Precisely, we enote by M the map obtaine from M by replacing R C by a single marke bounary face (of egree 2), an we enote by M 2 the map obtaine from M by replacing L C by a single marke internal face (of egree 2). It is clear that M is in A (2), while M 2 is in B (2a). Conversely, if we glue the marke bounary face of a map N A (2) to the marke internal face of a reuce map N 2 B (2a), we obtain a map M A (2a) whose maximal blocking cycle is the contour of the glue faces, so that N = M an N 2 = M 2. In orer to make the preceing ecomposition bijective, it is convenient to work with roote maps. Given a map M in A (2a), we efine M an M 2 as above, except that we mark a corner in the newly create bounary face of M. In orer to fix a convention, we choose the corner of M such that the vertices incient to the marke corners of M an M 2 are in the same block of the bipartition of the vertices of M. The ecomposition M (M, M 2 ) is now bijective an we call it the canonical ecomposition of the maps in A (2a). We summarize the above iscussion: Lemma 3.4. For all a, the canonical ecomposition is a bijection between A (2a) an A (2) B (2a). Note that the case a = above is special in that the set B (2) contains only the map {ɛ}. A similar strategy was alreay use in [2, 3, 4].
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES M A (2) C f s insie C C outsie C C M A (4) M 2 B (4) f s (a) (b) (c) () Figure 7. (a) A map in A (4) : the marke bounary face is the outer face, the marke internal face is inicate by a square, an the maximal blocking cycle C is rawn in bol. (b) The maps M an M 2 resulting from cutting M along C, each represente as a plane map enowe with its canonical biorientation (the marke inner face in each case is inicate by a square). (c) The mobiles associate to M an M 2. () The reuce bounary mobiles associate to M an M 2, where the marke vertex (corresponing to the marke inner face) is inicate by a square. Next, we escribe bijections for maps in A (2) an B (2a) by using a master bijection approach illustrate in Figure 7(b)-(). For M A (2), we call -orientation of M a consistent Z-biorientation of M with weights in {, 0, } such that: every internal ege has weight 0, every internal vertex has inegree, every non-marke internal face (of egree 4) has weight, while the marke internal face (of egree 2) has weight 0, every non-marke bounary of length 2i has weight (an inegree) i +, while the marke bounary (of length 2) has weight (an inegree) 0. Proposition 3.5. Let M be a map in A (2) consiere as a plane map by taking the outer face to be the marke bounary face. Then M amits a unique -orientation in Ô 2. We call it the canonical biorientation of M. Proof. This is a corollary of Proposition 3.. Inee, seeing M as a map D in D where an ege e is opene into an internal face f of egree 2, the canonical biorientation of M is irectly erive from the canonical biorientation of D, using the rules shown in Figure 8. For M A (2a), we call -orientation of M a consistent Z-biorientation with weights in {, 0, } such that:
2 O. BERNARDI AND É. FUSY 0 0 0 0 0 f f f 0 0 0-0 - 0 0 0 Figure 8. Transferring the biorientations an weights when blowing an ege into an internal face of egree 2. every internal ege has weight 0, every internal vertex has weight (an inegree), every non-marke internal face (of egree 4) has weight, while the marke internal face (of egree 2) has weight 0, every non-marke bounary of length 2i has weight (an inegree) i +, while the marke bounary (of length 2a) has weight (an inegree) a. Proposition 3.6. Let M be a map in A (2a) consiere as a plane map by taking the outer face to be the marke internal face. Then M has a -orientation in Ô2 if an only if it is reuce (i.e., is in B (2a) ). In this case, M has a unique -orientation in Ô2. We call it the canonical biorientation of M. The proof of Proposition 3.6 is elaye to Section 7. We enote by U the set of mobiles corresponing to (canonically oriente) maps in A (2) via the master bijection. By Theorem 2.2, these are the bounary Z-mobiles with weights in {, 0, } satisfying the following properties (which imply that the excess is 2): every ege has weight 0 (hence, is either black-black of weights (0, 0), or black-white of weights (, )), every black vertex has egree 4 an weight (hence has a unique white neighbor), except for a unique black vertex of egree 2 an weight 0, for all i 0, every white vertex of egree i + carries 2i legs. We also enote U the set of mobiles obtaine from mobiles in U by marking one of the corners of the black vertex of egree 2. For a, we enote by V (2a) the set of mobiles corresponing to (canonically oriente) maps in B (2a). These are the bounary Z-mobiles with weights in {, 0, } satisfying the following properties (which imply that the excess is 2): every ege has weight 0 (hence is either black-black of weights (0, 0), or black-white of weights (, )), every black vertex has egree 4 an weight (hence has a unique white neighbor), there is a marke white vertex of egree a which carries 2a legs, for all i 0, every non-marke white vertex of egree i + carries 2i legs. We also enote V (2a) the set of roote mobiles obtaine from from mobiles in V (2a) by marking one of the 2a legs of the marke white vertex. Propositions 3.5 an 3.6 together with the master bijection (Theorem 2.2) an Lemma 3.4 finally give: Theorem 3.7. The set A (2) (resp. A (2) Z-mobiles. Similarly, for all a, the set B (2a) set V (2a) (resp. V (2a) ) of Z-mobiles. ) of quarangulations is in bijection with the set U (resp. U ) of (resp. B (2a) ) of quarangulations is in bijection with the
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 3 Finally, the set A (2a) of quarangulations is in bijection with the set U V (2a) of pairs of Z-mobiles. The bijection is such that if the map M correspons to the pair of Z-mobiles (U, V ), then each non-marke bounary of length 2i in M correspons to a non-marke white vertex of U V of weight (an egree) i +, an each internal vertex of M correspons to a non-marke white leaf of U V. Theorem 3.7 is illustrate in Figure 7. 4. Bijections for triangulations with bounaries In this section we aapt the strategy of Section 3 to triangulations with bounaries, that is, maps with bounaries such that every internal face has egree 3. We start with the simpler case where one of the bounaries has egree before treating the general case. 4.. Triangulations with at least one bounary of length. Let D be the set of triangulations with bounaries, with a marke bounary face of egree. We think of maps in D as plane maps by taking the the marke bounary as the outer face. For M D, we call -orientation of M a consistent Z-biorientation with weights in { 2,, 0, } an with the following properties: every internal ege has weight (i.e., is either 0-way of weights (, 0), or -way of weights ( 2, )), every internal vertex has weight, every internal face has weight 2, every inner bounary of length i has weight (an inegree) i+, an the outer bounary has weight 0. Similarly as in Section 3. we have the following proposition prove in Section 7. Proposition 4.. Every M D has a unique -orientation in Ô. We call it the canonical biorientation of M. We enote by T the set of mobiles corresponing to (canonically oriente) maps in D via the master bijection. By Theorem 2.2, these are the bounary Z-mobiles satisfying the following properties (which reaily imply that the weights are in { 2,, 0, }, an the excess is ): every ege has weight (hence is either black-black of weights (, 0), or is black-white of weights ( 2, )), every black vertex has egree 3 an weight 2, for all i 0, every white vertex of egree i + carries i legs. To summarize, we obtain the following bijection for triangulations with a bounary of length (see Figure 9 for an example): Theorem 4.2. The set D is in bijection with the set T via the master bijection. If M D an T T are associate by the bijection, then each inner bounary of length i in M correspons to a white vertex in T of egree i +, an each internal vertex of M correspons to a white leaf in T. 4.2. Triangulations with arbitrary bounary lengths. We now aapt the approach of Section 3.2 (ecomposing maps into two pieces) to triangulations. For a, we enote by D (a) the set of triangulations with bounaries with a marke bounary face of egree a. We enote by D (a) the set of maps obtaine from maps in D (a) by also marking an arbitrary half-ege (either bounary or internal). We enote by A (a) the set of maps with bounaries having a marke bounary face of egree a an a marke internal face of egree, such that all the non-marke internal faces have egree 3. Lastly, we enote A (a) obtaine from A (2a) by marking a corner in the marke bounary face. the set of maps Given a map M in D (a), we obtain a map M in A (a) by the operation illustrate in Figure 0, which we call half-ege-opening. In wors, we open the ege containing the marke half-ege h into a face f, an
4 O. BERNARDI AND É. FUSY (a) (b) (c) Figure 9. (a) A triangulation in D enowe with its canonical biorientation, where crosses inicate half-eges of weight (-way eges have weights (, 2) if internal an weights (0, ) if bounary, 0-way eges have weights (, 0)). (b) The triangulation superimpose with the corresponing mobile. (c) The reuce bounary mobile (with i legs at each white vertex of egree i +, an with again the convention that half-eges of weight are inicate by a cross). then at the corner of f corresponing to h we insert a loop bouning the marke internal face (of egree ). This operation is clearly a bijection for a > : Lemma 4.3. For all a >, the half-ege-opening is a bijection between D (a) number of internal vertices an the bounary lengths. an A (a) which preserves the Note however that A () contains a map λ with eges (a loop separating a bounary an an internal face) which is not obtaine from a map in D (), so that the bijection is between D () an A () \ {λ}. v h u h u=v Figure 0. The operation of opening an half-ege h: it yiels a new face of egree surroune by a new face of egree 3 (the case where h is on a loop is illustrate on the right).
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 5 Next, we escribe a canonical ecomposition of maps in A (a) illustrate in Figure (a)-(b). For a cycle C of a map M A (a), we enote by R C an L C the regions boune by C containing an not containing the marke bounary face f s respectively. The cycle C is sai to be blocking if C has length (that is, is a loop), the marke internal face is in L C, an any bounary face incient to a vertex of C is in R C. Note that the contour of the marke internal face is a blocking cycle. It is easy to see that there exists a unique blocking cycle C such that L C is maximal (that is, contains L C for any blocking cycle C ). We call C the maximal blocking cycle of M. The maximal blocking cycle is inicate in Figure (a). The map M is calle reuce if its maximal blocking cycle is the contour of the marke internal face, an we enote by B (a) an B (a) the subsets of A (a) an A (a) corresponing to reuce maps. M A (2) f s insie C C C outsie C M A (3) M 2 B (3) (a) (b) (c) () C Figure. (a) A map in A (3) : the marke bounary face is the outer face, the marke internal face is inicate by a square, an the maximal blocking cycle C is rawn in bol. (b) The maps M an M 2 resulting from cutting M along C, each represente as a plane map enowe with its canonical biorientation (the marke inner face in each case is inicate by a square, each irecte ege has weights ( 2, ) an each unirecte ege has weights (, 0), with a cross on the half-ege of weight ). (c) The mobiles associate to M an M 2. () The reuce bounary mobiles associate to M an M 2, where the marke vertex is represente by a square. Black-white eges have weights ( 2, ), an black-black eges have weights (, 0), with a cross on the half-ege of weight. We now consier the two maps obtaine from a map M in A (a) by cutting the sphere along the maximal blocking cycle C, as illustrate in Figure (b). Precisely, we enote by M the map obtaine from M by replacing R C by a single marke bounary face (of egree ), an we enote by M 2 the map obtaine from M by replacing L C by a single marke internal face (of egree ). It is clear that M is in A (), while
6 O. BERNARDI AND É. FUSY M 2 is in B (a). The ecomposition M (M, M 2 ) is bijective (both for roote an unroote maps because A () A () ), an we call it the canonical ecomposition of maps in A (a). We summarize: Lemma 4.4. For all a, the canonical ecomposition is a bijection between A (a) an A () B (a). Next, we escribe bijections for maps in A () an B (a) by using the master bijection approach, as illustrate in Figure (b)-(). For M A (), we call -orientation of M a consistent Z-biorientation of M with weights in { 2,, 0, } such that: every internal ege has weight, every internal vertex has weight (an inegree), every non-marke internal face (of egree 3) has weight 2, an the marke internal face (of egree ) has weight 0, every non-marke bounary of length i has weight (an inegree) i +, an the marke bounary has weight (an inegree) 0. The following result easily follows from Proposition 4. (similarly as Proposition 3.5 follows from Proposition 3.). Proposition 4.5. Let M be a map in A (), consiere as a plane map by taking the marke bounary face as the outer face. Then M amits a unique -orientation in Ô. We call it the canonical biorientation of M. that: For M A (a) we call -orientation of M a consistent Z-biorientation with weights in { 2,, 0, } such every internal ege has weight, every internal vertex has weight (an inegree), every internal inner face (of egree 3) has weight 2, an the internal outer face (of egree ) has weight 0. every non-marke bounary of length i has weight (an inegree) i +, while the marke bounary of length a, has weight (an inegree) a. Proposition 4.6. Let M be a map in A (a) consiere as a plane map by taking the outer face to be the marke internal face. Then M has a -orientation in Ô if an only if it is reuce (i.e., is in B (a) ). In this case, M has a unique -orientation in Ô, which we call the canonical biorientation of M. Again the proof is elaye to Section 7. We enote by U the set of mobiles corresponing to (canonically oriente) maps in A () via the master bijection. These are the bounary Z-mobiles with weights in { 2,, 0, } satisfying the following properties (which imply that the excess is ): every internal ege has weight (hence is either black-black of weights (, 0), or black-white of weights ( 2, )), every black vertex has egree 3 an weight 2, except for a unique black vertex of egree an weight 0, for all i 0, every white vertex of egree i + carries i legs. For a, we enote by V (a) the set of mobiles corresponing to (canonically oriente) maps in B (a). These are the bounary Z-mobiles with weights in { 2,, 0, } satisfying the following properties (which imply that the excess is ): every internal ege has weight every black vertex has egree 3 an weight 2,
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 7 there is a marke white vertex of egree a which carries a legs, for all i 0, every non-marke white vertex of egree i + carries i legs. We also enote V (a) the set of mobiles obtaine from from mobiles in V (a) by marking one of the a legs of the marke white vertex. Propositions 3.5 an 4.6 together with the master bijection (Theorem 2.2) an Lemma 4.4 finally give: Theorem 4.7. The set A () set B (a) set A (a) of triangulations is in bijection with the set U of Z-mobiles. For all a, the ) of triangulations is in bijection with the set V (a) (resp. V (a) ) of Z-mobiles. Finally the of triangulations is in bijection with the set U V (a) of pairs of Z-mobiles. The bijection is such (resp. B (a) that if the map M correspons to the pair of Z-mobiles (U, V ), then each non-marke bounary of length i in M correspons to a non-marke white vertex in U V of weight (an egree) i +, an each internal vertex of M correspons to a non-marke white leaf in U V. Theorem 4.7 is illustrate in Figure. 5. Counting results 5.. Proof of Theorem.2 for quarangulations with bounaries. We efine a plante mobile of quarangulate type as a tree P obtaine as one of the two connecte components after cutting a mobile T T in the mile of an ege e; the half-ege h of e that belongs to P is calle the root half-ege of P, an the vertex incient to h is calle the root-vertex of P. The root-weight of P is the weight of h in T. For j {, 0, }, let R j R j (t; z 0, z, z 2,...) be the generating function of plante mobiles of quarangulate type having root-weight j, where t is conjugate to the number of bus, an z i is conjugate to the number of white vertices of egree i + (with 2i aitional legs) for i 0. We also enote R := t + R 0. The ecomposition of plante trees at the root easily implies that the series {R, R 0, R } are etermine by the following system (3) R = R 3, R 0 = 3R R 2, R = i 0 z ( 3i ) i i i R, where (for instance) the factor ( ) 3i i in the 3r line accounts for the number of ways to place the 2i legs when the root-vertex has egree i + (the root half-ege plus i chilren), an the factor 3 in the secon line accounts for choosing which of the 3 chilren of the root-vertex is white. This gives R = t + 3 i 0 z ( 3i ) i i R 3i+2, or equivalently, ( (4) R = tφ(r), with φ(y) = 3 ( ) 3i z i y 3i+). i i 0 Let U U (t; z 0, z,...) be the generating function of mobiles in U with t conjugate to the number of bus an z i conjugate to the number of white vertices of egree i + for i 0. For a, let V (2a) V (2a) (t; z 0, z,...) be the generating function of mobiles in V (2a) with t conjugate to the number of bus an z i conjugate to the number of non-marke white vertices of egree i + for i 0. The ecomposition at the marke vertex gives ( ) ( ) 3a 2 3a 2 U = R 2, an V (2a) = R a = R 3a 3. a a A (2a) (z 0, z,...) be the generating function of A (2a), where z 0 is conjugate to the number Let A (2a) of internal vertices an for all i, z i is conjugate to the number of unmarke bounaries of length 2i.
8 O. BERNARDI AND É. FUSY Theorem 3.7 gives A (2a) = U V (2a) t= = ( ) 3a 2 R 3a t=. a Now let β a (m; n,..., n h ) be the number of maps in D (2a) with a marke corner in the marke bounary face, with m internal vertices, n i non-marke bounaries of length 2i for i h, an no inner bounary of length larger than 2h. The half total bounary length is b = a + i in i, the total number of bounaries is r = + i n i. Moreover, by the Euler relation, the number of eges is e = 3b + 2r + 2m 4 = 3b + 2k, where k := r + m 2. Then Lemma 3.3 yiels e β a (m; n,..., n h ) = [z0 m z n zn h h ] A (2a) = ( ) 3a 2 [z0 m z n a zn h h ]R3a t=. It is easy to see from (4) that the variable t is reunant in R, an that for all q, n 0,... n h, [z n0 0 zn zn h h ]Rq t= = [z n0 0 zn zn h h ][tq+ h i=0 (3i+)ni ]R q. Moreover, by the Lagrange inversion formula [5, Thm 5.4.2], (4) implies that for any positive integers n, q, [t n ]R q = q n [yn q ]φ(y) n. Thus, enoting p := 3a + m + h i= (3i + )n i = m + r + 3b 2 = k + 3b, we get [z0 m z n zn h h ]R3a t= = [z0 m z n... zn h ][tp ]R 3a h = 3a ( [z0 m z n h h p ][yp 3a+ ] 3 = 3a [z0 m z n h h ( p ] 3 h i=0 z i ( 3i i = 3a ( p + m + r 3 m+r p p, m, n,..., n h h i=0 ) ) p Using k = m + r 2, p = k + 3b, e = p + k, an (3a ) ( ) 3a 2 a = 2 3 a( ) 3a a, we get ( (5) β a (m; n, n 2,..., n h ) = 3 k (e )! 3a m!(k + 3b)! 2a a ) h i= ( ) 3i z i y 3i+) p i ) h i= ( ) ni 3i, n i! i ( ) ni 3i. i which, multiplie by h i= n i!(2i) ni (to account for numbering the inner bounary faces an marking a corner in each of these faces), gives (2). 5.2. Proof of Theorem. for triangulations with bounaries. We procee similarly as in Section 5.. We call plante mobile of triangulate type any tree P equal to one of the two connecte components obtaine from some T T by cutting an ege e in its mile; the half-ege h of e belonging to P is calle the root half-ege of P, an the weight of h in T is calle the root-weight of P. For j { 2,, 0, }, let S j S j (t; z 0, z,...) be the generating function of plante mobiles of triangulate type an root-weight j, with t conjugate to the number of bus an z i conjugate to the number of white vertices of egree i + for
BIJECTIONS FOR PLANAR MAPS WITH BOUNDARIES 9 i 0. We also efine S := t + S. The ecomposition of plante trees at the root easily implies that the series {S 2, S, S 0, S } are etermine by the following system: (6) S 2 = S 2, S = 2SS 0, S 0 = 2SS + S0, 2 S = i 0 z i ( 2i i ) S 2 i. The secon line gives S 0 = 2 ( t S ). Hence the thir line gives S2 = t 2 + 8S S 3. Moreover the first an fourth line gives S = ( ) 2i z i S 2i. Thus, i i 0 (7) S = tφ(s), where φ(y) = ( 8 ( ) 2i z i y 2i+) /2. i i 0 Let U U (t; z 0, z,...) be the generating function of mobiles from U, with t conjugate to the number of bus an z i conjugate to the number of white vertices of egree i + for i 0. An for a let V (a) V (a) (t; z 0, z,...) be the generating function of mobiles in V (a), with t conjugate to the number of bus an z i conjugate to the number of non-marke white vertices of egree i+ for i 0. A ecomposition at the marke vertex gives Let A (a) A (a) U = S, an V (a) = ( ) 2a 2 S a 2 = a ( ) 2a 2 S 2a 2. a (z 0, z,...) be the generating function of A (a), where z 0 is conjugate to the number of internal vertices an for all i, z i is conjugate to the number unmarke bounaries of length i. Theorem 4.7 gives (8) A (a) = U B (a) = ( ) 2a 2 S 2a t=. a We efine now η a (m; n, n 2,..., n h ) as the number of triangulations with a marke bounary of length a having a marke corner, with m internal vertices, n i non-marke bounaries of length n i for i h, an no non-marke bounary of length larger than h. The total bounary-length is b := a + i in i, the number of bounaries is r = + i n i, an (by the Euler relation) the number of eges is e = 2b + 3r + 3m 6, which is 2b + 3k with k := r + m 2. Then Lemma 4.3 yiels 2e η a (m; n, n 2,..., n h ) = [z0 m z n zn h ] A (a) It is easy to see from (7) that for all positive integers q, n 0,..., n h, [z n0 0 zn zn h h h ]Sq t= = [z n0 0 zn ( ) = [z0 m z n zn h 2a 2 h ] S 2a t=. a zn h h ][tq+ h i=0 (2i+)ni ]S q.
20 O. BERNARDI AND É. FUSY Hence, by the Lagrange inversion formula, an using the notation p := 2a +m+ h i= (2i+)n i = 2b+k an s = m + i n i = k + gives [z0 m z n... zn h h ]S2a t= = [t p z0 m z n... zn h h ]S2a = 2a [y p 2a+ z0 m z n p... zn h h ( ] 8 = 2a [z0 m z n p... zn h h ( ] 8 = 2a [z0 m z n p... zn h h ] = 2a p 8 s ( = 2a 4 s( h m! i= s ( m, n,..., n h n i! ( ) 2i ni) i Thus, using p + 2s 2 = e, s = k +, an p = 2b + k we get ( η a (m; n, n 2,..., n h ) = 4 k (e 2)!! 2a m! (2b + k)!! a a h i=0 z i ( 2i i h ( ) 2i z i y 2i+) p/2 i i=0 ) ) p/2, h ( ) ) s 2i 8 z i [u s ]( u) p/2 i i=0 ) ( h ( ) 2i ni) (p + 2s 2)!! i (p 2)!!s!2 s i= (p + 2s 2)!!. p!! ) h i= ( ) ni 2i. n i! i Multiplying this expression by h i= n i!i ni (to account for numbering the inner bounary faces an marking a corner in each of these faces) gives (). 5.3. Solution of the imer moel on quarangulations an triangulations. A imer-configuration on a map M is a subset X of the non-loop eges of M such that every vertex of M is incient to at most one ege in X. The eges of X are calle imers, an the vertices not incient to a imer are calle free. The partition function of the imer moel on a class B of maps is the generating function of maps in B enowe with a imer configuration, counte accoring to the number of imers an free vertices. The partition function of the imer moel is known for roote 4-valent maps [6, 5] (an more generally p-valent maps). We observe that counting (roote) maps with imer configurations is a special case of counting (roote) maps with bounaries. More precisely, upon blowing each imer into a bounary face of egree 2, a roote map with a imer-configuration can be seen as a roote map with bounaries, such that all bounaries have length 2, an the roote corner is in an internal face. Base on this observation we easily obtain from Theorem.2 that, for all m, r 0 with m + 2r 3, the number q m,r of imer-configurations on roote quarangulations with r imers an m + 2r vertices is (9) q m,r = 4(m + 2r 2) 32r+m 2 (5r + 2m 5)!. r!m!(4r + m 2)! Similarly, Theorem. implies that, for all m, r 0 with m+2r 3, the number t m,r of imer-configurations on roote triangulations with r imers an m + 2r vertices is (0) t m,r = (m + 2r 2) 22m+3r 3 3 r+ (7r + 3m 8)!!. r!m!(5r + m 2)!!