ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 225-266 https://doi.org/10.26493/1855-3974.1115.90f (Also available at http://amc-journal.eu) Enumeration of hypermaps of a given genus* Alain Giorgettif FEMTO-ST Institute, Univ. Bourgogne Franche-Comté, CNRS 16 route de Gray, 25030 Besancon cedex, France Timothy R. S. Walsh Department of Computer Science, University of Quebec in Montreal (UQAM) P. O. Box 8888, Station A, Montreal, Quebec, Canada, HC3-3P8 Received 23 May 2016, accepted 17 October 2017, published online 20 June 2018 This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces. Keywords: Enumeration, surface, genus, rooted hypermap, unrooted hypermap. Math. Subj. Class.: 05C30, 05A15 1 Introduction A (combinatorial) hypermap is a triple (D,R,L) where D is a finite set of darts and R and L are permutations on D such that the group (R, L) generated by R and L acts transitively on D. A (combinatorial ordinary) map is a hypermap (D, R, L) whose permutation L is a fixed-point-free involution on D. For a hypermap (resp. map) the orbits of R, L and * The authors wish to thank Alexander Mednykh, Roman Nedela and the referees for helpful suggestions to improve the presentation of this article. ^For this work Alain Giorgetti was supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03). E-mail addresses: alain.giorgetti@femto-st.fr (Alain Giorgetti), walsh.timothy@uqam.ca (Timothy R. S. Walsh) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ 226 Ars Math. Contemp. 15 (2018) 147-160 RL (L followed by R) are respectively called vertices, hyperedges (resp. edges) and faces. The degree of a vertex, edge, hyperedge or face is the number of darts it contains. The equivalence of combinatorial maps and topological maps having been established in [14], we use the word "map" to mean "combinatorial map" throughout this paper. The genus g of a map is given by the Euler-Poincare formula [7] v - e + f = 2(1 - g), (1.1) where v is the number of vertices, e is the number of edges and f is the number of faces. The genus of a hypermap with t darts, v vertices, e hyperedges and f faces was defined in [13] by the formula v + e + f = t + 2(1 - g). (1.2) An isomorphism between two maps or hypermaps (D, R, L) and (D', R', L') is a bi-jection from D onto D' that takes R into R' and L into L'; it corresponds to an orientation-preserving homeomorphism between two topological maps. A sensed hypermap (resp. map) is an isomorphism class of hypermaps (resp. maps). We admit the existence of a unique hypermap (resp. map) with an empty set of darts D, called the empty hypermap (resp. map). For both of these objects v = f =1 and g = e = 0. A rooted hypermap (resp. map) is either the empty hypermap (resp. map) or a tuple (D, x, R, L) where (D, R, L) is a non-empty combinatorial hypermap (resp. map) and x G D is a distinguished dart, called the root. The enumeration of maps and hypermaps has several non-trivial applications. One such application is based on the correspondence between hypermaps and algebraic curves established by the Belyi theorem [16]. For instance, the formula for the number of plane trees was used by A. Zvonkin in the computer generation of Shabat polynomials of bounded degree [16]. Another area where the map enumeration plays an important role is theoretical physics, in particular in 2-dimensional gravitation models. Roughly speaking, map enumeration is used to compute matrix integrals determining the properties of gravitational fields (see for instance the works of B. Eynard [9]). Some hypermaps have been shown to be related to contextuality in quantum physics [21]. Also, A. Mednykh and R. Nedela have applied the enumeration of rooted (resp. unrooted) hypermaps to the enumeration of subgroups (resp. conjugacy classes of subgroups) of the triangle group with three generators x, y, z and the relation xyz = 1 [20]. We enumerate rooted hypermaps of a given genus by number of darts, vertices, hyper-edges and faces. To do so we consider more general families of rooted hypermaps and bipartite maps, in which other vertices or darts than the root dart are distinguished. We also use the genus-preserving bijection between hypermaps and 2-vertex-coloured bipartite maps presented in [23]. But since bipartite maps have all their faces of even degree and we're using the degrees of the vertices as parameters, we must instead study the face-vertex dual of a 2-coloured bipartite map, that is, a map whose faces are coloured in two colours (white and black) so that no two faces that share an edge have the same colour. All these maps are Eulerian - that is, all their vertices are of even degree - but not all Eule-rian maps are 2-face-colourable. For example, the map on the torus with one vertex, one face and two edges is Eulerian because its only vertex is of degree 4, but its face cannot be coloured because it shares both edges with itself. Therefore we call the maps we are studying face-bipartite. A sequenced (rooted) map is a rooted map with some vertices other than the root vertex (the vertex that contains the root) distinguished from each other and from all the other A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 227 vertices. The labels that distinguish these vertices can be taken to be 1, 2,..., k, where k is the number of distinguished vertices. A sequenced (rooted) hypermap is defined similarly. We state (in Section 4) a bijective decomposition for the set H(g, t, f, e, n, D) of sequenced orientable hypermaps of genus g with t darts, f faces and e hyperedges, with the root vertex of degree n and with the sequence of degrees of the distinguished vertices equal to D = (di, d2,..., d|D|), where dj is the degree of the distinguished vertex with label i. We obtain a bijective decomposition of the set F(g, e, w, b, n, D) of sequenced orientable face-bipartite maps of genus g with e edges, w white faces, b black faces, with the root face of degree 2n and with the sequence of half-degrees of the distinguished vertices equal to D. Then we apply face-vertex duality to obtain a bijective decomposition of the corresponding set of 2-coloured bipartite maps with distinguished faces. Next we use the bijection in [23] to obtain a bijective decomposition for hypermaps with distinguished faces, and finally we again apply face-vertex duality to obtain a bijective decomposition of H(g, t, f, e, n, D). A mutirooted hypermap is a hypermap in which a non-empty sequence of darts with pairwise distinct initial vertices is distinguished. We relate multirooted hypermaps to se-quenced hypermaps and thus obtain a recurrence for the number of multirooted hypermaps and functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the initial vertices of the distinguished darts. The paper is organized as follows. Section 2 fixes some notations, recalls a known decomposition for sequenced rooted maps and describes the bijection between hypermaps and bipartite maps presented in [23]. Sections 3 and 4 respectively enumerate sequenced face-bipartite maps and sequenced rooted hypermaps of a given genus. In Section 5 we consider multirooted hypermaps and we give equations for the generating functions that count these objects. In Section 6 we give functional equations relating the generating functions for rooted hypermaps with that for multirooted hypermaps. Then we show how to solve these equations. In Section 7 we obtain parametric expressions for the generating functions that count rooted hypermaps with a given small positive genus. Section 8 presents enumeration algorithms for sensed unrooted hypermaps counted by number of darts, vertices and hyperedges. Appendix A (resp. B) contains a table for numbers of rooted (resp. unrooted) hypermaps of genus g with d darts, v vertices and e hyperedges for d < 14. 2 Background 2.1 Notations We first introduce the notations and conventions we use throughout the paper. Let D and D' be two lists of integers. The inclusion D' C D means that D' is a sublist of D. In this case D - D' is the complementary sublist of D' in D. For instance, the sublists of D = [1,1,2] are the empty list [], [1] (twice), [2], [1,1], [1,2] (twice) and D itself. Their complementary sublists in the same order are D, [1, 2] (twice), [1,1], [2], [1] (twice) and []. We denote by D.D' the concatenation of the lists D and D'. If i is an integer and D is a list of integers, then i.D is a shortcut for [i].D. For 1 < j < |D| we denote by dj the j-th element of the list D of length |D| and by D - {dj} the list obtained from D by removing its j-th element dj. Let p be a positive integer. The abbreviation D1..p denotes the list [d1,..., dp]. The abbreviation Vd p denotes v-1 ... vpp. The sign + (resp. denotes (resp. generalized) disjoint set union in the following decompositions and (resp. generalized) arithmetic sum in the following equations. By con- 228 Ars Math. Contemp. 15 (2018) 147-160 vention, a disjoint set union (resp. sum) over an empty domain is equal to the empty set (resp. zero). For any logical formula p the notation Av means the singleton set containing only the empty hypermap or map (depending on the context) and the empty set if p is false. The notation 5V means 1 if p is true and 0 if p is false. 2.2 Bijective decomposition of the set of sequenced maps In 1962 W. T. Tutte [22] presented a bijective decomposition of a planar map with all the vertices distinguished and a root in every vertex. In 1972 T. R. Walsh and A. B. Lehman [27] generalized this decomposition to maps of higher genus and used it to count rooted maps of a given genus by number of vertices and faces. In 1987 D. Arques [3] used this latter decomposition to find a closed-form formula for the number of rooted maps of genus 1 by number of vertices and faces. In 1991 E. A. Bender and E. A. Canfield [4] presented a more efficient decomposition that roots only a single vertex and distinguishes only as many other vertices as necessary and used it to obtain explicit formulas for counting rooted maps of genus 2 and 3. In 1998 the first author [11] modified this decomposition and used it to obtain a bijective decomposition satisfied by the set M(g, e, f, n, D) of sequenced orientable maps of genus g with e edges and f faces, with the root vertex of degree n and with D the list of degrees of the distinguished vertices was obtained in [11]. Since this bijective decomposition contains an error, we present the correct bijective decomposition here, and we derive it to make the derivation more accessible than the contents of a Ph. D. thesis. Theorem 2.1. The set M(g, e, f, n, D) of sequenced orientable maps of genus g with e edges and f faces, with the root vertex of degree n and with the list D of degrees of the distinguished vertices is defined by the bijective decomposition M(g, e, f, n, D) = M(gi,ei,fi,ni,Di) x M(g2, e2, f2,n2, D - Di) gi + 92 = g ei + e2 = e — i fi + f2 = f ni + n2 = n — 2 Di C D n—3 + ^ M(g - 1, e - 1, f,n - 2 - p,p.D) x {1,...,p} (2.1) p=i p=2e—2 + ^ M(g,e - 1, f,p, D) p=n—i |D| + M(g,e - 1, f, dj + n - 2,D - {dj}) + A(gie,f,n,D)=(0,0,i,0,[]). j=i Proof. If a map m has at least one edge, we reduce by 1 the number of edges by the face-vertex dual of deleting the root edge. There are two cases of this operation, depending upon whether the root edge is a loop or a link, and each of these cases breaks down into two sub-cases. Case 1: The root edge is a loop. We delete the root edge and split the root vertex into two parts, si and s2. If r is the root, then si consists of the darts R(r), R2(r),..., R—i(L(r)) A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 229 and s2 consists of the darts R(L(r)), R2(L(r)), ..., R x(r). This case breaks down into two cases, depending upon whether or not this operation disconnects the map. Case 1a: This operation disconnects the map into two maps, m1 containing s1 and m2 containing s2 .If m1 has at least 1 edge, its root is r1 = R(r), and if m2 has at least 1 edge, its root is r2 = R(L(r)). Let g1, e1, f1, n1, D1 and g2, e2, f2, n2, D2 be the parameters of the maps m1 and m2, respectively, corresponding to g, e, f, n, D. This operation reduces by 1 the total number of edges; so e1 + e2 = e - 1. It leaves unchanged the total number of faces because r and L(r) simply get deleted from the cycle(s) of RL (L followed by R) containing them; so f1 + f2 = f. It increases by 1 the total number of vertices; so from Formula (1.1), which relates the genus of a map to the number of its vertices, faces and edges, it can easily be deduced that g1 + g2 = g. It decreases by 2 the total number of darts in s1 and s2 since r and L(r), which belonged to the root vertex, get eliminated; so n1 + n2 = n - 2. Finally, D1 can be any sublist of D and D2 is just the complementary sublist, denoted by D - D1. This operation is uniquely reversible; so the set of ordered pairs of sequenced maps obtained in this case is ^ M(g1,e1,f1 ,n1,D1) x M(g2, e2, f2, n2, D - D1), (2.2) gi + 32 = g ei + e2 = e — 1 fl + f2 = f n i + n 2 = n — 2 Di C D where E means the union of disjoint sets. Case 1b: This operation does not disconnect the map, but instead turns it into a new map m' with e - 1 edges and f faces and, since the number of vertices increases by 1, the genus of m' is g - 1, so that this case only occurs when g > 1. Neither s1 nor s2 can be of degree 0 (otherwise the map would be disconnected); so we can choose for m' the root r1 = R(r) belonging to s1. Let p be the degree of s2. Since the sum of the degrees of s1 and s2 is n - 2, the degree of s1, the root vertex, is n - 2 - p. We distinguish the vertex s2 so that this operation can be reversed, and we put its degree p at the beginning of the list D, turning it into p.D. Now this operation is reversible in p distinct ways, since any of the p darts of s2 can be chosen to be R(L(r)) when we merge the vertices s1 and s2 and replace the deleted root edge. Now p can be any integer from 1 up to n - 3 (so that n - 2 - p > 1). For both p and n - 2 - p to be at least 1, n must be at least 4. The set of sequenced maps obtained in this case is n—3 ^ M(g - 1, e - 1, f, n - 2 - p,p.D) x {1,...,p}. (2.3) p=1 Case 2: The root edge is a link. We contract the root edge, merging its two incident vertices s1 containing the root r and s2 containing L(r) into a single vertex s with root R(r). This operation decreases by 1 the number of edges and doesn't change the number of faces, since r and L(r) simply get deleted from the cycle(s) containing them. Since the number of vertices is decreased by 1, the genus remains the same. This case breaks down into two sub-cases, depending upon whether or not s2 is one of the distinguished vertices. Case 2a: The vertex s2 is not one of the distinguished vertices. Let p be the degree of the new vertex s. Then p = n - 2 + the degree of s2, and since the degree of s2 must be 230 Ars Math. Contemp. 15 (2018) 147-160 at least 1, we have p > n - 1. Also, the new map has 2e - 2 darts; so p < 2e - 2. This operation is uniquely reversible for each value of p; so the set of maps so obtained is p=2e-2 ^ M(g,e - 1,f,p,D). (2.4) p=n-1 Case 2b: The vertex s2 is one of the distinguished vertices. It can be any one of the |D| distinguished vertices. If it is the jth distinguished vertex, then its degree is dj. Then since it gets merged with s1 into the new root vertex, dj gets dropped from D. Finally, the degree of s is dj + n - 2. This operation too is uniquely reversible; so the set of maps so obtained is |D| ^ M(g, e - 1, f, dj + n - 2, D - {dj}). (2.5) j=1 Finally, suppose that m has no edges. It is of genus 0, has 1 face, its one vertex is of degree 0 and its list D is empty because it has no distinguished vertices; so it constitutes the singleton A(s,e,/,n,D) = (0,0,1,0,[]). (2.6) Then M (g, e, f, n, D) is the disjoint union of the sets given by (2.2) - (2.6). □ 2.3 Bipartite maps and hypermaps To motivate the transformation of (2.2)-(2.6) into the corresponding equations for sequenced hypermaps we briefly describe the bijection in [23] that takes a hypermap h into a 2-coloured bipartite map m = I(h), its incidence map. The bijection I takes the darts, vertices and hyperedges of h into the edges, white vertices and black vertices of m. A root (distinguished dart) of h corresponds to a distinguished edge of m; to make it correspond to a root of m we impose the condition that a root of m belongs to a white vertex. The permutation R in h corresponds to R in m acting on a dart in a white vertex and the permutation L in h corresponds to R in m acting on a dart in a black vertex. The permutation L in m doesn't correspond to any permutation in h; rather, since it takes a dart belonging to a vertex of one colour into a dart belonging to a vertex of the opposite colour, it toggles R in m between R and L in h. A face (cycle of RL) in h corresponds to a face in m with twice the degree. To see this, we follow one application of RL in h starting with a dart d, which corresponds to an edge in m but we make it correspond to the dart d' in that edge that also belongs to a white vertex. Then the L in h takes d' first into L(d'), which belongs to a black vertex, and then into RL(d') and the following R in h takes RL(d') first into LRL(d'), which belongs to a white vertex, and then into RLRL(d'). Since the genus of a hypermap with t darts, v vertices, e hyperedges and f faces is defined by (1.2), m has the same genus as h. Since the root of an incidence map of a rooted hypermap must belong to a white vertex, we impose the condition on a rooted 2-face-coloured face-bipartite map that the root belong to a white face and we transform (2.2)-(2.6) into the corresponding bijective decomposition for these maps. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 231 3 Sequenced face-bipartite maps Let F(g, e, w, b, n, D) be the set of sequenced orientable face-bipartite maps of genus g with e edges, w white faces, b black faces, with the root face of degree 2n and with the list of half-degrees of the distinguished vertices equal to D. For any dart d we denote by f (d) the face containing d and we note that the face f (R(d)) = f (L(d)) must have the opposite colour from f (d) because those two faces share the edge {d, L(d)}. Theorem 3.1. The set F(g, e, w, b, n, D) satisfies the bijective decomposition F(g, e, w, b, n, D) = ^ F(gi,ei,wi,bi,ni,Di) x F(g2, e2, w2, b2, n2, D - D1) g i + 92 = g ei + e2 = e — i wi + b2 = b W2 + bi = w n i + n2 = n — i Di C D n—2 + F(g — 1, e — 1, b, w, n — 1 — p,p.D) x {1,... ,p} (3.1) p=i p=e— i + E F(g, e — 1, b, w,p, D) p= n |D| + ^F(g, e — 1, b, w, dj + n — 1, D — {dj}) + A(gie,w,b,n,D)=(0,0,i,0,0,[]). j=i Proof. Case 1: The root edge is a loop. By definition, f (r), where r is the root of the map m, is white, so that since ri = R(r), f (ri) must be black. But when the loop is removed and the vertex s containing r is split, ri becomes a root; so f (ri) must change colour and so must all the faces of the new map m' (in case 1b) or the map mi containing ri (in case 1a). In case 1a, the other map m2 has r2 = RL(r) as a root and f (r2) is white; so its faces stay the same colour. This implies that in case 1a wi + b2 = b and w2 + bi = w, whereas in case 1b w and b switch in going from m to m'. In case 1a, we have, as for general maps, gi + g2 = g, ei + e2 = e — 1 and Di is any subset of D, but instead of ni + n2 = n — 2 we have ni + n2 = n — 1 because the degrees satisfy the equation 2ni + 2n2 = 2n — 2. The analogue of formula (2.2) is thus T: F(gi, ei, wi, bi, ni, Di) xF(g2, e2, w2, b2, n2, D — Di). (3.2) 9i + 92 = 9 ei + e2 = e — i wi + b2 = b W2 + bi = w ni + n2 = n — i Di C D In case 1b, the reduced map m' is still of genus g — 1 and has e — 1 edges, but the degree of s2 is now 2p instead of p and the degree of the new root vertex si is 2(n — 1 — p); so the parameter n — 2 — p in (2.3) changes to n — 1 — p. Also, 1 < 2p < 2n — 3, but since 2p is even, we have 1 < p < n — 2 instead of 1 < p < n — 3, and the condition that n > 4 232 Ars Math. Contemp. 15 (2018) 147-160 changes to n > 3. The analogue of formula (2.3) is thus n-2 F(g — 1, e — 1, b, w, n — 1 — p,p.D) x {1,... ,p}. (3.3) p=i Case 2: The root edge is a link. Since the new root R(r) belongs to a black face, all the faces change colour; so b and w switch. In case 2a, we have 2n — 1 < 2p < 2e — 2, but since 2p is even, we now have n < p < e — 1; so the analogue of (2.4) is p=e-1 ^ F(g, e — 1,b,w,p,D). (3.4) p=n In case 2b, the degree of the new root vertex is 2dj + 2n — 2; so the analogue of (2.5) is |D| ^F(g, e — 1,b, w,dj + n — 1, D — {dj}). (3.5) j=i Finally, the map with no edges has one white face and no black ones; so the analogue of (2.6) is ^(g,e,w,6,n,D) = (0,0,1,0,0,[]). (3.6) □ After deriving this bijective decomposition, we became aware of the article [8], which presents a similar bijective decomposition but for multi-rooted face-bipartite maps, which are like sequenced face-bipartite maps except that every distinguished vertex has a root. However, we present our derivation here for several reasons: it makes our article self-contained, we obtained it independently of [8] and our main purpose is to count hypermaps rather than face-bipartite maps. Now [8] does present a construction that converts a hy-permap into a face-bipartite map. However, that construction is not proved and it is far more complicated than the one in [23], which is not cited in [8]. We also recently became aware of the article [6], which generalizes the results of [15] by computing the generating functions for edge-labelled bipartite maps on an orientable surface of genus g with an unbounded number of faces and including the degrees of these faces as parameters. 4 Sequenced rooted hypermaps Theorem 3.1 holds for rooted 2-coloured bipartite maps with distinguished faces, where e is the number of edges, w is the number of white vertices, b is the number of black vertices, n is half the degree of the root face and D is the list of half-degrees of the distinguished faces. By the bijection described in Section 2.3, it also holds for rooted hypermaps with distinguished faces, where e is the number of darts, w is the number of vertices, b is the number of hyperedges, n is the degree of the root face and D is the list of degrees of the distinguished faces. By duality, the theorem also holds for sequenced hypermaps, where e is the number of darts, w is the number of faces, b is the number of hyperedges, n is the degree of the root vertex and D is the list of degrees of the distinguished vertices. To make the letters correspond to the objects they represent, we change F to H, e to t, w to f and b to e. We thus obtain the following results. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 233 Theorem 4.1 (Bijective decomposition for sequenced hypermaps). Let H(g, t, f, e, n, D) be the set of sequenced orientable hypermaps of genus g with t darts, f faces and e hy-peredges, with the root vertex of degree n and with the list of degrees of the distinguished vertices equal to D = (d\,d2,... ,d\D\), where di is the degree of the distinguished vertex with label i. The set H(g, t, f, e, n, D) satisfies the bijective decomposition H(g,t,f,e,n,D) = E H(gi,ti,fi,ei,ni,Di) x H(g2,t2, f2,e2,n2, D - Di) g i + 92 = g tl + t2 = t — i fl + e2 = e f2 + ei = f ni + n 2 = n — 1 D1 C D n—2 + E H(g - l,t - 1, e, f, n - 1 - P,P.D) x{l,...,p] (4.1) p=i p=t—i + E H(g,t - 1, e, f,P, D) p= n |D| + E H(g,t - 1 e f,dj + n - 1,D - {dj}) + A(g,t,f,e,n,D) = (0,0,l,0,0,[]). j=i Corollary 4.2 (Recurrence between numbers of sequenced hypermaps). Let H(g, t, f, e, n, D) be the number of rooted sequenced hypermaps of genus g with t darts, f faces and e hyperedges such that the root vertex is of degree n and D is the list of degrees of the distinguished vertices. Then H(0,0,1,0,0, []) = 1 and if t > 1, then H(g,t,f, e, n, D) = E H(gi,ti,fi,ei,ni,Di) H(g2,t2, f2,e2,n2, D - Di) 91 + 92 = 9 ti + t2 = t — 1 fl + e2 = e f2 + ei = f n i + n2 = n — 1 Di C D n—2 + ¿n>3^g>i E PH (g - 1,t - 1, e, f, n - 1 - p,p.D) (4.2) p=i p=t—i + E H(g,t - 1, e, f,p, D) p= n |D| + E H(g, t - 1,e, f, dj + n - 1,D - {dj}). j=i 5 Multirooted hypermaps For p > 1 a p-rooted hypermap is a hypermap in which a sequence of p darts with pairwise distinct initial vertices is distinguished. A multirooted hypermap is a p-rooted hypermap 234 Ars Math. Contemp. 15 (2018) 147-160 for some p > 1. This section addresses the enumeration of multirooted hypermaps. Theorem 5.1 (Recurrence between numbers of multirooted hypermaps). Let Hm (g,t,f,e, D) be the number of multirooted hypermaps of genus g with t darts, f faces and e hyper-edges such that D is the list of degrees of the distinguished vertices. Then Hm(0,0,1,0, [ ]) = 1 and if t > 1, then Hm(g,t, f, e, n.D) = Hm(gi,t1 ,fi,ei,ni.Di) Hm(g2 ,h, f2,e2,n2.(D - Di)) g i + g 2 = g ti + t2 = t — i fl + e2 = e f2 + ei = f ni + n2 = n — i Di C D n—2 + Sn>sSg>^ Hm(g - 1,t - 1,e, f, (n - 1 - p).p.D) (5.1) p=i p=t—i + ^ Hm(g,t - 1, e, f,p.D) p= n |D| + ^ dj Hm(g,t - 1, e, f, (dj + n - 1).(D - {dj})). j=i Proof. A multirooted hypermap is similar to a sequenced rooted hypermap except that for each distinguished non-root vertex a dart starting from it is distinguished. If the degree of the j th distinguished vertex is dj, then there are dj ways of distinguishing a dart of this vertex. It follows that for each sequenced rooted hypermap, there are njD}idj multirooted hypermaps. Let Hm(g, t, f, e, D) be the number of multirooted hypermaps of genus g with t darts, f faces and e hyperedges such that such that D is the list of degrees of the initial vertex of the distinguished darts. Then Hm(g, t, f, e, n.D) = H(g, t, f, e, n, D) ngdj. (5.2) Solving (5.2) for H(g, t, f, e, n, D) and substituting into (4.2) proves the theorem. □ For p > 1 let Hg (vi, .. . ,vp,x,y,u,z) = ^ Hm(g,t,f,e,Di..p)vDipp xf yeuv zf (5.3) t > 0, f > i, e > 0 di > i, . . . , dp > i = t + 2(i - g) - e - f be the generating function that counts multirooted hypermaps of genus g with p distinguished darts if g > 0, and 0 otherwise. For 1 < i < p, the exponent d^ of the variable vi in this series is the degree of the initial vertex of the i-th distinguished dart. The exponent f of the variable x is the number of faces, the exponent e of the variable y is the number of hyperedges, the exponent t of the variable z is the number of darts and the exponent v of the variable u is the number of vertices (v is computable from the other parameters by Formula (1.2)). A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 235 Corollary 5.2 (Functional equations for multirooted hypermaps). For g > 0 and p > 1 the generating functions Hg of multirooted hypermaps of genus g are defined by the following functional equations: Hg (vi,W,x,y,u,z) = g yviz xu *—' '—' j ^ ^ Hj (v1,X,y,x,u,z)Hg-j (vi,W — X,x,y,u,z) j=0 XCW V\z + — Hg-i(vi,vi, W, y, x, u, z) (5.4) V\u,z vi — 1 j=p + --7 (Hg(vi,W,y,x,u,z) — Hg(1,W,y,x,u, z)) d ( Hg (vj ,W — {vj },y,x,u,z) — Hg (vi,W — {vj },y,x,u,z) Ed I ±±g(Uj ,=_2dj (v'--vt — vi + xuSg=oSp=i, where W = v2,... ,vp. Proof. By summation according to (5.3) of the recurrence between numbers of multirooted hypermaps from Theorem 5.1. □ By vertex-hyperedge duality, we have Hg(vvi, W,y,x,u, z) = Hg(vi,W,x,y,u, z) + Sg=oSp=i(yu — xu) (5.5) and thus another functional equation without x, y swaps is: Hg (vi,W,x,y,u,z) = v g / —£ [(Hj(v1,X,x,y,u,z) + Sj=oS\x\=o(yu — xu)) xu j=0 XCW V Hg-j (v1,W — X, x, y, u, z) v1z + — Hg-i(vi,vi,W,x,y,u,z) (5.6) v1uz H--7 (Hg (vi, W, x, y, u, z) — Hg (1,W,x,y,u, z)) v1 — 1 j=P a +vivzYJvj d^w j=2 + xuSg = oSp=l. d f Hg (vj ,W — {vj },x,y,u,z) — Hg (vi,W — {vj },x,y,u,z) dvj vj vj — v! The former equation is given here for maximal generality. However, a consequence of the genus formula (1.2) is that three variables among the four variables x, y, u and z are sufficient. In the remainder of the paper we consider the generating functions Hg (vi,W,x,y,u) = Hg (vi,W,x,y,u, 1) 236 Ars Math. Contemp. 15 (2018) 147-160 with one fewer variable. They are defined by the following functional equations: Hg (vi, W,x, y, u) = g -E E (Hj(vi,X,x,y,u) + ¿j,oJ|x|,o(yM - xu)) Hg_j(vi,W - X,x,y,u) j=0 XCff vi + — Hg_i(vi,wi, W,x,y, u) (5.7) + (Hg(vi, W,x,y,u) - Hg(1, W,x,y,u)) vi — 1 + j=p A Hg(vj, W - {vj},x,y,u) - Hg(vi, W - {vj},x,y,u) +j=v dvj ^ - vi + x«ig = 0ip=i. For g, p = 0,1, after grouping in the left-hand side the terms containing Hg (vi, W, x, y, u) in (5.7), one gets A(vi,x,y,u) -Hg (vi,W,x,y,u) = vi g x(1 - vi) ^^ ^^ Hj(vi, X, x, y, u)Hg_j(vi,W - X, x, y, u) j=0 x c W (j, X) = (0, []) (j, X) = (g, W) 1 — vi +--u— Hg_i(vi, vi, W, x, y, u) + uHg(1, W, x, y, u) + wTg(vi,W, x, y, u) (5.8) A(v, x, y, u) = vu + (1 - v)(1 - yv + xv - 2vHo(v, x, y, w)/w) (5.9) Tg(vi, W,x,y,u) = with and 3=P d / v / (1 - vi) E v- dv- ( v— (v-,W - {v-xu) -=2 - Hg (vi,W - {v- },x,y,u)JJ . (5.10) 6 Rooted hypermap generating functions Let hg (v, e, f ) be the number of rooted genus-g hypermaps with v vertices, e hyperedges and f faces. Let Hg (x,y,u) = E hg (v,e,f )xv ye«/ (6.1) v,e,f >1 be the ordinary generating function for counting rooted hypermaps on the orientable surface of genus g > 0, where the exponent of variable x is the number of vertices, the exponent of variable y is the number of hyperedges, and the exponent of variable u is the number of faces. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 237 Rooted hypermaps being 1-rooted hypermaps, Hg (x,y,u) = Hg (1,x,y,u), (6.2) where Hg (v\,... ,vp,x, y, u) is the generating function counting p-rooted genus-g hypermaps defined in Section 5 for p > 1. We first recall in Section 6.1 a known parametric expression of the generating function that counts rooted planar hypermaps. Then we explain in Section 6.2 how to solve the functional equation of the generating functions Hg (x, y, u) that count rooted hypermaps with a given positive genus g. 6.1 Rooted planar hypermaps The following proposition is a reformulation of [1, Theorem 3], with the correspondence s = x, f = u and a = y for variables, A = p, p = q and v = r for parameters, and Ho = sf (1 + J) for generating functions. Proposition 6.1 ([1]). The ordinary generating function H0(x,y,u) that counts rooted planar hypermaps by number of vertices (exponent of x), hyperedges (exponent of y) and faces (exponent of u) is the unique solution of the following parametric system: H0(x,y,u) = 1 + pqr(1 — p — q — r) (6.3) with = p(1 - q - r) = q(1 — p — r) (6.4) ky = r(1 — p — q). Proof. The generating function H0(v, x, y, u) that counts rooted planar hypermaps (genus 0) by number of vertices (exponent of x), hyperedges (exponent of y), faces (exponent of u) and degree of the root vertex (exponent of v) satisfies the functional equation yv Ho(v,x,y,u) = —(Ho(v, x,y,u) + yu — xu) Ho(v,x,y,u) xu vu +--- (Ho(v, x, y, u) — Ho(1, x, y, u)) + xu (6.5) v—1 obtained by instantiation of (5.7) with g = 0, p =1 and v\ = v. This equation can be solved by the quadratic method [10, page 515]. The idea is to define auxiliary functions A(v, x, y, u) and B(v, x, y, u) by (5.9) and B (v,x,y,u) = A(v,x,y,u)2 (6.6) and look for a function V(x, y, u) such that A(V (x,y,u),x,y,u) = 0, (6.7) implying that B(V(x, y, u), x, y,u) = 0 and dvB(v, x, y, v,)\v=V(XyVuU) = 0. We get from (6.5), (5.9) and (6.6) that B(v, x, y, u) = 1 — 2yv — 2xv — 2v3y — 2v3x — 2v2u + v4y2 — 2v3y2 + y2v2 + v4 x2 o o 9 9 9 9 9 9 A — 2v x + x v + v u + 4v yx — 2yv x — 2yv u + 2v yu — 2v yx — 2v3xu + 2xv2u + 4v2x + 4v2y + 2vu + 4v3H0(1, x, y, u) — 4v2H0(1,x,y,u) — 2v + v2. (6.8) 238 Ars Math. Contemp. 15 (2018) 147-160 The constraints B(V(x, y, u), x, y, u) = 0 and B(v, x, y, u)|v=V(x,y,u) = 0 respectively are 1 - 2yV - 2xV - 2V3y - 2V3 x - 2V2u + V4y2 - 2V3y2 + y2V2 + V4x2 - 2V3x2 + x2V2 + V2u2 + 4V3yx - 2yV2x - 2yV2u + 2V3yux - 2V4y - 2V3xu + 2xV2u + 4V2x + 4V2y + 2Vu + 4V3H0(1,x,y,u) - 4V2H0(1,x,y,u) - 2V + V2 = 0 (6.9) and -2 + 8yV + 8xV + 4V3y2 - 6y2V2 + 4V3x2 - 6x2V2 - 6V2x - 6V2y - 4Vu + 2y2 V + 2x2V + 2Vu2 - 4yVu + 4xVu - 4yVx + 12yV2x + 6yV2u - 8V3yx - 6xV2u + 2V - 2x - 2y + 2u = 0. (6.10) It can be checked that both equations are satisfied by V =1/(1 - q) with x, u, y and H0(1, x, y, u) related to p, q and r by (6.4) and (6.3). (6.11) □ 6.2 Rooted hypermaps with positive genus The following additional notations are used in this section. Let p be a positive integer. Let Hj[ni,..., np] denote the partial derivative of the function Hj(vi,..., vp, x, y, u) with respect to the variables v1,..., vp to the respective orders n1,..., np, computed at v1 = ... = vp = V. The abbreviation [p] denotes the list [2,..., p] if p > 2 and the empty list [] if p =1. The abbreviation N[p] denotes the list [n2,..., np]. For any sublist X C [p] of [p], [p] - X denotes the sublist of the elements of [p] that are not in X, NX denotes the list of those n in N[p] such that i is in X and Nj denotes the list [n2,..., nj_1, nj+1,..., np]. 6.2.1 Equation for rooted hypermaps and recurrence relations The special case of Formula (5.8) for g > 1, p =1 and v1 = V is the following formula: uHg (1, x, y, u) = I '-1 \ (V - 1) I x^Hj(V, x, y,u)Hg-j(V,x,y, u) + HS_1(V, V, x,y,u)/u I i.e. uHg(1, x, y, u) = (V - 1) |x E Hj[0]Hg_j[0] + Hg_1[0, 0]/u| . (6.12) In order to derive from (6.12) a value for Hg(1, x, y, u), we are looking for a value for Hj[0], Hg_j[0] and Hg-1[0,0]. More generally, we will derive from the following proposition a closed form for the expressions Hg [n1,..., np]. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 239 Proposition 6.2. For g > 0, p > 1 and ni,..., np > 0 the function Hg [ni,..., np] is defined by (ni + 1)A[1] V H g [ni,N[p]] i+j+k=n i + 1 V ,J 7 i>0, k 1, M(v) = 1 — v and L(v) = v(1 — v). Proof. Equation (6.13) is obtained from Equation (5.8) as follows: 1. Partial derivation of (5.8) with respect to the variables vi, v2,..., vp to the respective orders ni + 1, n2, ..., np. 2. Evaluation of this differential equation at vi = • • • = vp = V. The function Hg [ni + 1,..., np] is multiplied by A[0] in the resulting equation, and A[0] is known to be zero (6.7). The functions Tg [...] are replaced by expressions with the functions Fg [...] thanks to Lemma 6.3 below. 3. In the left-hand side of the resulting equation, isolation of the single term involving the function Hg[ni,..., np]. By inspection one can check that the right-hand side of (6.13) depends only on some functions Hg[k, n2,..., np] with k < ni, some functions Hg [ni,..., np,] with p' < p and some functions Hj [...] for j < g. Thus, (6.13) in a recursive definition of the family of functions Hg [ni,..., np] for g > 0, p > 1 and ni,..., np > 0. □ The following lemma relates the partial derivatives of Tg at v = V with the ones of Fg. Lemma 6.3. For p > 2 and g, ni,..., np > 0, Tg [ni + 1,N[p] = E (ni + 1)!nj ! (njFg [ni + nj + 2, Nj] ^ (ni + n + 2)! V j g[ i + j + ' j] + i) x j=2 (i ' ) + S++3Fg [ni + nj + 3, Nj]). (6.15) 240 Ars Math. Contemp. 15 (2018) 147-160 Proof. We can easily prove that d I"(vj - vi)Hs(vi, [p] - {vj},x, y,u) dvj vj - vi Then, Tg (v1,..., vp, x, y, u) equals (6.16) vj j=2 (vj - vi) 1 (vj(1 - vi)Hg(vj, [P] - {vj},x,y,u) - vi(1 - vi)Hg (vi, [p] -{vj },x, y,u)) . (6.17) It also holds that ni + i dni dvni+i vj (vj - vi)Hg (vj, [P] - {vj},x,y,u) vj - vi so that d rt^+fi Tg (v1,..., vp, x, y, u) equals (6.18) j=r d n 1 +2 j=fj dvn 1 + idvAV j - i (vj - vi) i (vj(1 - vj)Hg(vj, [p] - {vj}, x, y, u) - vi(1 - vi)Hfl (vi, [p] -{vj },x,y,u) (6.19) i.e. j=p j=2 dvni + i dvj dni+2 (Fg(vj, [p] -{vj},x,y,u) - Fg(vi, [p] - {vj},x,y,u) vj - vi Formula (6.15) is a consequence of dni+n2 f ^(x1) - ^(x2) \ dx^1dx^2 V X1 - X2 / ni!n2! xi=x2 = a (ni + n2 + 1)! The formula Fg [n,N] = £ (n)L[k]Hg [l, N] fc+i=n ^ ' (6.20) ^(ni +n2+i)(a). (6.21) □ (6.22) is an easy consequence of (6.14). Thus the right-hand side of (6.13) only depends on some functions Hg [k,..., np] with k < n1, some functions Hg [n1,..., n'p,] with p' < p, some functions Hj[...] for j < g and some functions A[i]. A relation between A[i] and some functions H0[j] is established in Section 6.2.2. 6.2.2 Case g = 0 and p =1 The function A[i] can be related to some functions H0[j] as follows: With M(v) = 1 - v and L(v) = v(1 - v), Equation (5.9) is A(v, x, y, u) = vu + M (v) + L(v)(-y + x - 2xHo(v, x, y, u)). (6.23) 0 0 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 241 Its instantiation at v = V gives 1 - q Ho[0] = --. (6.24) 1 — q — r For k > 1, the k-th partial derivative of (6.23) in v is d k Qk Qk ^ A(v,x,y,u) = dVk (vu) + dVk M (v) dk + dVk [L(v)(-y + x - 2xHo(v, x, y, u))] (6.25) and its instantiation in v = V is d k A[k] = dV^ (vu)|v=v + M [k] + ^ (k)L[i] (dj(-y+x - 2xHo(v,x,y,u))iv=^. (6.26) Solving (6.26) for k = 1 gives (1 - q)2 (A[1] + 1 - p - q - r) Ho[1] = ^-^ 2 (1 -^-". (6.27) 2pq(1 - q - r) For k > 2, one gets i+j=k since M [k] = 0, i.e. A[k] = -2x £ (k)i[i]Ho[j] i i j=k \ / A[k] = -2x ^L[0]Fo[k] + kL[1]Ho[k - 1] + k(k2 1) L[2]Ho[k - 2] ) (6.28) since L[k] = 0 if k > 3. 7 Explicit formulas for small genera This section proposes explicit parametric expressions for the generating functions that count rooted hypermaps of small positive genus. In Section 7.1 we count by number of vertices, hyperedges and faces; the number of darts can be obtained from these parameters by Formula (1.2). In Section 7.2 we count by number of darts alone. 7.1 Rooted hypermap series enumerated with three parameters For g = 1,..., 5 we have computed an explicit expression of Hg (x, y, u) parameterized by p, q and r, with x = p(1 - q - r), u = q(1 -p - r) and y = r(1 -p - q), by application of formulas in Section 6. For g > 3, the expressions are too large to be included in the present text, but a Maple file with all the generating functions up to genus 5 is available from the first author on request. 242 Ars Math. Contemp. 15 (2018) 147-160 A parametric expression of Hi (x, y, u) is p q r (1 — p) (1 — q) (1 — r) Hi(x,y,u) = (7.1) [(1 — p — q — r)2 — 4pqr]2 This expression can be derived from [2, Theorem 3], with the correspondence s = x, f = u, and a = y between variables and the correspondence Hi(x, y, u) = xuKi(1, x, y, u) between generating functions. A parametric expression of H2 (x, y, u) is where H2(x,y,u) = p q r (1 — p) (1 — q) (1 — r) P2(p, q,r) [(1 — p — q — r)2 — 4pqr]7 (7.2) P2(p, q, r) = 76p6q2r2 — 8p4q4r2 — 8p4q2r4 + 76p2q6r2 — 8p2q4r4 + 76p2q2r6 + 40p7qr — 76p6q2r — 76p6qr2 — 112p5q3 r — 228p5q2r2 — 112p5qr3 + 8p4q4r + 16p4q3r2 + 16p4q2r3 + 8p4qr4 — 112p3 q5r + 16p3q4r2 + 40p3q3r3 + 16p3q2r4 — 112p3qr5 — 76p2q6r — 228p2q5r2 + 16p2q4r3 + 16p2q3r4 — 228p2q2r5 — 76p2qr6 + 40pq7r — 76pq6r2 — 112pq5r3 + 8pq4r4 — 112pq3r5 — 76pq2 r6 + 40pqr7 + p8 — 20p7q — 20p7r — 35p6q2 — 64p6qr — 35p6r2 + 56p5q3 + 396p5q2r + 396p5qr2 44 + 56p5r3 + 140p4q4 + 264p4q3r + 393p4q2r2 + 264p4qr3 + 140p4r + 56p3q5 + 264p3q4r — 92p3q3r2 — 92p3q2r3 + 264p3qr4 + 56p3r5 — 35p2q6 + 396p2 q5r + 393p2q4r2 — 92p2q3r3 + 393p2q2r4 + 396p2qr5 — 35p2r6 — 20pq7 — 64pq6r + 396pq5r2 + 264pq4r3 + 264pq3r4 + 396pq2r5 — 64pqr6 — 20pr7 + q8 — 20q7r — 35q6 r2 + 56q5r3 + 140q4r4 + 56q3r5 — 35q2r6 — 20qr7 + r8 + 6p7 + 105p6q + 105p6r + 21p5q2 — 116p5qr + 21p5r2 — 420p4q3 — 821p4q2 r — 821p4qr2 34 — 420p4r3 — 420p3q4 — 648p3q3r — 316p3q2 r2 — 648p3 qr3 — 420p3r + 21p2q5 — 821p2 q4r — 316p2q3r2 — 316p2q2r3 — 821p2qr4 + 21p2r5 + 105pq6 — 116pq5r — 821pq4r2 — 648pq3r3 — 821pq2r4 — 116pqr5 25 + 105pr6 + 6q' + 105q6r + 21q5r2 — 420q4r3 — 420q3r4 + 21q2r + 105qr6 + 6r7 — 49p6 — 189p5q — 189p5r + 315p4q2 + 479p4qr + 315p4r2 + 910p3q3 + 1162p3q2r + 1162p3qr2 + 910p3r3 + 315p2q4 + 1162p2q3r + 720p2q2r2 + 1162p2qr3 + 315p2r4 — 189pq5 + 479pq4r + 1162pq3r2 + 1162pq2r3 + 479pqr4 — 189pr5 — 49q6 — 189q5r + 315q4r2 + 910q3r3 + 315q2r4 — 189qr5 — 49r6 + 112p5 + 70p4q + 70p4r — 770p3q2 — 876p3qr — 770p3r2 — 770p2q3 — 1380p2q2r — 1380p2qr2 — 770p2r3 + 70pq4 — 876pq3r — 1380pq2r2 — 876pqr + 70pr4 + 112q5 + 70q4r — 770q3r2 — 770q2r3 + 70qr4 + 112r5 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 243 - 105p4 + 210p3q + 210p3r + 735p2q2 + 1034p2 qr + 735p2r2 + 210pq3 + 1034pq2r + 1034pqr2 + 210pr3 - 105q4 + 210q3r + 735q2r2 + 210qr3 - 105r4 + 14p3 - 315p2q - 315p2r - 315pq2 - 672pqr - 315pr2 + 14q3 - 315q2r - 315qr2 + 14r3 + 49p2 + 175pq + 175pr + 49q2 + 175qr + 49r2 - 36p - 36q - 36r + 8. Remark: For g = 0, the formula H0(x, y,u) = pqr(1 - p - q - r) (7.3) can be derived from [1], with the correspondence s = x, f = u, and a = y between variables and the correspondence H0(x, y, u) = xuK0(1,x,y,u) between generating functions. 7.2 Rooted hypermap series enumerated by number of darts Let Hg (z) be the ordinary generating function of rooted hypermaps on the orientable surface of genus g > 0, where the exponent of variable z is the number d of darts. 7.2.1 Generating functions For g from 0 to 6, a parametric expression of Hg (z), where z = t(1 - 2t) and t = 0 when z = 0, is Ho(z) t3 (1 - 3 t ) z2 , (7.4) Hi(z) t 3 (1 - t) (1 - 4 t)2 , (7.5) H2(z) 4 z2 t3 (51 t4 - 77 t3 + 48 t2 - 15 t + 2) (1 - t)5 (1 - 4 t)7 , (7.6) H3(z) 4 z4 t3 P3(z) = (1 - t)9 (1 - 4 t)12 , (7.7) H4(z) 4 z6 t3 P4(z) (1 - t )13 (1 - 4 t )17 , (7.8) H5(z) 4 z8 t3 P5(z) (1 - t)17 (1 - 4 t)22 , (7.9) H6(z) 4 z10 t3P6(z) = (1 - t)21 (1 - 4 T)27 , (7.10) with P3 (z) = 28496 t9 - 36888 t8 - 13164 t7 + 61676 t6 - 61872 t5 + 35172 t4 - 13168 t3 + 3360 t2 - 552 t + 45, 244 Ars Math. Contemp. 15 (2018) 147-160 P4(z) = 32375616 t 14 + 15509760 t13 — 243313744 t12 + 442844592 t 11 — 389268768 t 10 + 170357328 t9 + 1281984 t8 — 53553072 t7 + 39814032 t6 — 17597520 t5 + 5541192 t4 — 1320920 t3 + 239697 t2 — 30456 t + 2016, P5(z) = 61742404608 t 19 + 239043447552 t 18 — 1163002515456 t 17 + 1403096348736 t 16 + 338393916800 t 15 — 2962590413376 t 14 + 4243997599488 t 13 — 3552865706240 t 12 + 2000782619136 t 11 — 761565230016 t10 + 165542511744 t9 + 7568059872 t8 — 23295865824 t7 + 11016156244 t6 — 3336459144 t5 + 761835465 t4 — 141393220 t3 + 21738240 t2 — 2490480 t + 151200 P6 (z) = 178054771302400 t24 + 1584534210564096 t23 — 4933663711730688 t22 and 6 — 2073822560019456 t21 + 28025505345377280 t20 — 55010184951564288 t 19 + 54283457920223232 t 18 — 22997164994372352 t 17 — 13439214645718272 t 16 + 31734000656779264 t 15 — 29719458122609664 t 14 + 18704646148809216 t 13 — 8736443315384448 t 12 + 3098312828500416 t 11 — 813298324826016 t 10 + 138473163256176 t9 — 4043551301232 t8 — 6580517850696 t7 + 2630924485729 t6 — 626336383104 t5 + 112079088144 t4 — 17314508592 t3 + 2485496880 t2 — 284717376 t + 17107200. We have also computed the generating functions for 7 < g < 11. Their expressions are too large to be included in the present text, but a Maple file is available from the first author on request. A. Mednykh and R. Nedela used our formulas (7.4) to (7.7) to find explicit formulas for the number of rooted hypermaps for genus g = 0,1, 2 and 3 [19]. 7.3 Other parameterization In a private communication to the second author, P. Zograf suggests the parameterization t z =(TT27. a11) After adding the condition that t = 0 when z = 0, it corresponds to 1 — 4z — V1 — 8z , t =-^-. (7.12) 8z These two parameterizations are equivalent. The one can be transformed into the other by means of the following substitutions: t (7.13) 1 + 2t A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 245 and 4 = . (7.14) By means of these substitutions, the following parametric expressions in t can be obtained from the parametric expressions (7.4)-(7.10) for Hg (t) in t: Ho(z) Hi(z) H2(z) Hs(z) = t (1 - t), (1+ t)(1 - 2t)2 ' 4 t5 (1 + 2t) (t4 - t3 + 6 t2 +1 + 2) (1+1)5(1 - 2t)7 , 4 t7 (1 + 2t) (1 +t)-9(1 - 2t)-12 (80 t9 - 120 t8 + 1500 t7 + 1036 t6 + 3768 t5 + 2820 t4 + 2288 t3 + 1008 t2 + 258 t + 45), H4(z) = 4 t9 (1 + 2t) (1 +t)-13(1 - 2t) 7 (16768 t14 - 33536 t ,13 + 653776 t12 + 786480 t11 4358016 t10 + 6151056 t9 + 10059552 t8 H5(z) H6(z) For 0 Moreover. + 10217040 t7 + 8418240 t6 + 5227024 t5 + 2365888 t4 + 800128 t3 + 181665 t2 + 25992 t + 2016), = 4 t11 (1 + 2t) (1 + t)-17(1 - 2t)-22 ( 67 3 2800 t19 - 16832000 t18 + 450011520 t17 + 773106240 t16 + 5764983552 t15 + 11910647232 t14 + 29130502912 t13 + 46090300928 t12 + 63452543616 t11 + 68713116608 t10 + 60654218080 t9 + 43591208976 t8 + 25142796864 t7 + 11637842232 t6 + 4232899206 t5 + 1181820745 t4 + 245635580 t3 + 35501760 t2 + 3255120 t + 151200), = 4 t13 (1 + 2t) (1 + t)-21(1 - 2t)-27 (4424052736 t24 - 13272158208 t23 + 452750478336 t22 + 1012254206976 t21 + 9488911137792 t20 + 25803592571904 t19 + 83891900050944 t18 + 180120643165440 t17 + 346626234587904 t16 + 535272874975232 t15 + 701152993531392 t14 + 771688966862592 t13 + 716686355273472 t12 + 563018634260736 t11 + 372549313187520 t10 + 207088794784752 t9 + 96021082581732 t8 + 36765061031004 t7 + 11475757049569 t6 + 2863185376896 t5 + 556090776432 t4 + 80913152016 t3 + 8274846384 t2 + 536428224 t + 17107200). < g < 3, these expressions correspond to Fg (t) in Zograf's communication. they reveal an extra factorization by 4(1 + 2t) for g > 2. 8 Efficient enumeration of rooted and sensed unrooted hypermaps by number of darts, vertices and hyperedges We recall that a sensed map or hypermap is an equivalence class of (unrooted) maps or hypermaps under orientation-preserving isomorphism. Before enumerating sensed hypermaps we first need to enumerate rooted hypermaps. We use an efficient method of counting rooted hypermaps by number of darts, faces, ver- 3 t 246 Ars Math. Contemp. 15 (2018) 147-160 tices and hyperedges or, equivalently [23], 2-coloured bipartite maps rooted at a white vertex by number of edges, faces, white vertices and black vertices, presented by Kazarian and Zograf [15], and then count sensed 2-coloured bipartite maps and hypermaps with the same parameters using the same method we used [26, 12] to count sensed maps by number of edges, faces and vertices. The recurrence (formula (11) in [15]), with f changed to H, is as follows. Define Hg,d to be a homogeneous polynomial in the three variables t, u, and v. The coefficient of tf ubvw in Hg,d is the number of 2-coloured bipartite maps of genus g with d edges, f faces, b black vertices and w white vertices rooted at a white vertex or, equivalently, the number of rooted hypermaps of genus g with d darts, f faces, b hyperedges and w vertices. Then H0ji = tuv and (d +1)Hg,d = (2d — 1)(t + u + v)Hg,d—i + (d — 2) (2(tu + tv + uv) — (t2 + u2 + v2)) Hg,d—2 (8.1) g d—3 + (d — 1)2 (d — 2)Hg—i,d—2 + EE(4 + 6J')(d — 2 — j)Hi,j Hg—2— j. i=0 j=i In [26] we collaborated with Mednykh to enumerate rooted and sensed maps. Med-nykh enumerated maps of genus up to 11 by number of edges alone, while we enumerated maps of genus up to 10 by number of edges and vertices. The method we used to enumerate rooted maps is presented in [25]. The method we used to enumerate sensed maps is based on Liskovets' refinement [17] of the method Mednykh and Nedela used to enumerate sensed map of genus up to 3 by number of edges [18]. Later we used a more efficient method of enumerating rooted maps, presented in [5], to enumerate rooted and sensed maps of genus up to 50 [12]. To describe here the modifications we made to pass from maps to 2-coloured bipartite maps we need to briefly discuss a few of the concepts described in more detail in [26]. All the automorphisms of a map on an orientable surface are periodic. If the period is L > 1, then the automorphism divides the map into L isomorphic copies of a smaller map, called the quotient map. Most of the cells (vertices, edges and faces) are in orbits of length L under the automorphism; those that aren't are called branch points. For example, if a map is drawn on the surface of a sphere which undergoes a rotation through 360/L degrees, the two cells through which the axis of rotation pass are fixed; so they are each in an orbit of length 1 for any L. For maps of higher genus, not all the branch points are on orbits of length 1. For example, if a torus is represented as a square with opposite edges identified in pairs, and is rotated by 90 degrees (period 4), then the centre of the square is a branch point of orbit length 1 and so is the point represented by all four corners of the square, but the middle of the sides of the square are two branch points of orbit length 2: the point represented by the middle of both vertical sides of the square is taken by the rotation onto the point represented by the middle of both horizontal sides, and vice versa; so it takes two rotations to take either of these points back onto itself. Also, if the middle of an edge is a branch point, then the quotient map contains half of that edge - a dangling semi-edge. An automorphism of a map M of genus G is characterized by the following parameters: the period L, the genus g of its quotient map and the number of branch points of each orbit length. If each orbit length is replaced by its branch index (L divided by the orbit length), we obtain what is called an orbifold signature in [18]. In [18] a method is presented for determining which orbifold signatures could characterize an automorphism A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 247 of a map of genus G (a G-admissible orbifold) and how many such automorphisms could be characterized by that orbifold signature; a variant of that method is presented in [17], and this is the one we use except that we deal with orbit lengths instead of branch indices. The method used in [18] to enumerate sensed maps of genus G with E edges by number of edges can be roughly described as follows. For each G-admissible orbifold O, let g be the genus of the quotient map, L be the period and qi be the number of branch points with branch index i. Then the number vo (d) of rooted maps with d darts that could serve as a quotient map for an automorphism with that signature once the branch points are pasted onto the map in all possible ways is given by VO (d) = E (")(("-')/2+2-2g)N, ((d - .)/2), (8.2) VV \q2-s,qs,...,qL J where Ng (n) is the number of rooted maps of genus g with n edges (0 if n is not an integer). Here s is the number of dangling semi-edges in the quotient map m, all of which must be in orbits of length L/2 so that they represent normal edges in the original map M. The binomial coefficient is the number of ways of inserting dangling semi-edges into the rooted map multiplied by d/(d - s) because there are d ways to root the map once the dangling edges have been inserted and only d- s ways to root it without the dangling edges. The multinomial coefficient is the number of ways to distribute the branch points with the various branch indices among the non-edges of the quotient map; the number at the top of the multinomial coefficient is the number of non-edges and is given by the Euler-Poincare formula (1.1). Then the number of sensed maps of genus G with E edges is 2E EEEpio(*I(0),zL) vO(2E/L), (8.3) L|E O where O runs over all the G-admissible orbifolds with period L and Epi0(ni(O), ZL) is the number of automorphisms that have the orbifold signature of O. In [26] we distributed the branch points that aren't on dangling semi-edges among the vertices and faces separately. The quotient map of a bipartite map can't contain any dangling semi-edges; otherwise the lifted map would have an edge joining two vertices of the same colour. Here we distribute the branch points among the white vertices, black vertices and faces, and, like in [26], we don't use a formula like (8.3); instead we compute the contribution of each orbifold signature to the number of sensed 2-coloured bipartite maps whose number of white vertices, black vertices, faces and edges are allowed to vary within a user-defined upper bound on the number of edges. Suppose that the quotient map is of genus g and has w white vertices, b black vertices and f faces. Then the number e of edges can be calculated from the formula f - e + w + b = 2(1 - g) (8.4) and the number d of darts is 2e. Suppose also that among the branch points of orbit length i, wi are on a white vertex, bi are on a black vertex and fi are in a face. We denote by wL, bL and fL the number of white vertices, black vertices and faces, respectively, that do not contain a branch point. The original map will have W white vertices, B black vertices and F faces, where L L L W = E iwi, B = E ibi and F = E ifi, i=1 i=1 i=1 (8.5) 248 Ars Math. Contemp. 15 (2018) 147-160 and the total number E of edges is equal to Le = F + W + B - 2(1 - g). The binomial coefficient in (8.2) disappears because the quotient map can't contain any dangling semi-edges. The multinomial coefficient must be replaced by the number of ways to distribute the branch points among the white vertices, black vertices and faces. Then (8.2) becomes vo (d, w, b, f) = ( w V, b b b Vf f f f Vfl(d,w,b,f), (8.6) where d is the number of edges in the quotient maps on both sides of the formula (or the number of darts in the corresponding hypermaps) and Ng (d, w, b, f) is the number of 2-coloured bipartite maps with d edges with w white vertices, b black vertices and f faces, rooted at a white vertex. For this number to be positive, the sum of all the w» cannot exceed w with a similar bound on the sum of all the bj and the sum of all the f»; so w, b and f each starts at its respective sum and increases by 1 until the number E of edges in the original map exceeds a user-defined maximum. With each increase of w, b or f, one of the multinomial coefficients in (8.6) gets updated using a single multiplication and division. The product of these three multinomial coefficients must be computed for all sets of nonnegative integers such that for each i, w» + bj + f is equal to the total number of branch points of orbit length i. Once (8.6) is multiplied by the number of automorphisms with the current orbifold signature, we get the contribution of that signature and the numbers w», b» and f to E times the number of sensed 2-coloured bipartite maps of genus G with E edges, F faces, B black vertices and W white vertices. This contribution is added to the appropriate element of an array, initially 0, and when all the contributions have been tallied, for each E, F, W and B the corresponding array element is divided by E (not 2E because the root must be incident to a white vertex) to give the number of sensed 2-coloured bipartite maps of genus G with E edges, F faces, B black vertices and W white vertices or, equivalently, the number of sensed hypermaps of genus G with E darts, F faces, B hyperedges and W vertices. This enumeration was done with a program written in C++ using CLN to treat big integers. It enumerated rooted and sensed hypermaps of genus up to 24 with up to 50 darts as fast as it could display the numbers on the screen. The numbers coincide with those obtained by generating the hypermaps [24]. The source code is available from the second author on request. References [1] D. Arques, Relations fonctionnelles et denombrement des hypercartes planaires pointees, in: G. Labelle and P. Leroux (eds.), Combinatoire enumerative, Springer, Berlin, volume 1234 of Lecture Notes in Mathematics, pp. 5-26, 1986, doi:10.1007/bfb0072505. [2] D. Arques, Hypercartes pointees sur le tore: Decompositions et denombrements, J. Comb. Theory Ser. B 43 (1987), 275-286, doi:10.1016/0095-8956(87)90003-7. [3] D. Arques, Relations fonctionnelles et denombrement des cartes pointees sur le tore, J. Comb. Theory Ser. B 43 (1987), 253-274, doi:10.1016/0095-8956(87)90002-5. [4] E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Comb. Theory Ser. B 53 (1991), 293-299, doi:10.1016/0095-8956(91)90079-y. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 249 [5] S. R. Carrell and G. Chapuy, Simple recurrence formulas to count maps on orientable surfaces, J. Comb. Theory Ser. A 133 (2015), 58-75, doi:10.1016/j.jcta.2015.01.005. [6] G. Chapuy and W. Fang, Generating functions of bipartite maps on orientable surfaces, Electron. J. Comb. 23 (2016), #P3.31, http://www.combinatorics.org/ojs/index. php/eljc/article/view/v2 3i3p31. [7] H. S. M. Coxeter, Regular Polytopes, Dover, New York, 3rd edition, 1973. [8] P. Dunin-Barkowski, N. Orantin, A. Popolitov and S. Shadrin, Combinatorics of loop equations for branched covers of sphere, Int. Math. Res. Notices (2017), rnx047, doi:10.1093/imrn/ rnx047. [9] B. Eynard, Formal matrix integrals and combinatorics of maps, in: J. Hamad (ed.), Random Matrices, Random Processes and Integrable Systems, Springer, New York, pp. 415-442, 2011, doi:10.1007/978-1-4419-9514-8_6. [10] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, New York, NY, USA, 1st edition, 2009, http://ac.cs.princeton.edu/home/. [11] A. Giorgetti, Combinatoire bijective et ¿num^rative des cartes pointees sur une surface, Ph.D. thesis, Universite de Marne-la-Vallee, Institut Gaspard Monge, 1998. [12] A. Giorgetti and T. R. S. Walsh, Constructing large tables of numbers of maps by orientable genus, 2014, arXiv:1405.0 615 [math.CO] . [13] A. Jacques, Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Paris Ser. A 267 (1968), 625-627. [14] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (1978), 273-307, doi:10.1112/plms/s3-37.2.273. [15] M. Kazarian and P. Zograf, Virasoro constraints and topological recursion for Grothendieck's dessin counting, Lett. Math. Phys. 105 (2015), 1057-1084, doi:10.1007/s11005-015-0771-0. [16] S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, volume 141 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, doi:10.1007/ 978-3-540-38361-1, see especially chapter 2, "Dessins d'Enfants", pp. 79-153. [17] V. A. Liskovets, A multivariate arithmetic function of combinatorial and topological significance, Integers 10 (2010), 155-177, doi:10.1515/integ.2010.012. [18] A. Mednykh and R. Nedela, Enumeration of unrooted maps of a given genus, J. Comb. Theory Ser. B 96 (2006), 706-729, doi:10.1016/j.jctb.2006.01.005. [19] A. Mednykh and R. Nedela, Counting hypermaps by Egorychev's method, Anal. Math. Phys. 6 (2016), 301-314, doi:10.1007/s13324-015-0119-z. [20] A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci. 226 (2017), 635-654, doi:10.1007/s10958-017-3555-5. [21] M. Planat, A. Giorgetti, F. Holweck and M. Saniga, Quantum contextual finite geometries from dessins d'enfants, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550067, doi:10.1142/ s021988781550067x. [22] W. T. Tutte, A census of slicings, Canad. J. Math. 14 (1962), 708-722, doi:10.4153/ cjm-1962-061-1. [23] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory Ser. B 18 (1975), 155-163, doi:10.1016/0095- 8956(75)90042- 8. [24] T. R. S. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps, J. Integer Seq. 18 (2015), Article 15.4.3, https://cs.uwaterloo.ca/journals/JIS/ VOL18/Walsh/walsh3.html. 250 Ars Math. Contemp. 15 (2018) 147-160 [25] T. R. S. Walsh and A. Giorgetti, Efficient enumeration of rooted maps of a given orientable genus by number of faces and vertices, Ars Math. Contemp. 7 (2014), 263-280, doi:10.26493/ 1855-3974.190.0ef. [26] T. R. S. Walsh, A. Giorgetti and A. Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), 2660-2671, doi: 10.1016/j.disc.2011.11.027. [27] T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus I, J. Comb. Theory Ser. B 13 (1972), 192-218, doi:10.1016/0095-8956(75)90050-7. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 251 A First numbers of rooted hypermaps The following sections show the numbers h of rooted hypermaps of genus g with d darts, v vertices, e edges and d — v — e + 2(1 — g) faces, for g < 6 and d < 14. A.1 Genus 0 d v e f h 6 1 5 2 15 8 2 5 3 2436 1 1 1 1 1 6 2 4 2 135 8 3 4 3 7500 6 3 3 2 262 8 4 3 3 7500 1 sum 1 6 4 2 2 135 8 5 2 3 2436 6 5 1 2 15 8 6 1 3 196 2 1 1 2 1 6 1 6 1 1 8 1 7 2 28 2 1 2 1 1 6 2 5 1 15 8 2 6 2 518 2 2 1 1 1 6 3 4 1 50 8 3 5 2 2436 6 4 3 1 50 8 4 4 2 3985 2 sum 3 6 5 2 1 15 8 5 3 2 2436 6 6 1 1 1 8 6 2 2 518 3 1 1 3 1 8 7 1 2 28 3 1 2 2 3 6 sum 1584 8 1 8 1 1 3 2 1 2 3 8 2 7 1 28 3 1 3 1 1 7 1 1 7 1 8 3 6 1 196 3 2 2 1 3 7 1 2 6 21 8 4 5 1 490 3 3 1 1 1 7 2 1 6 21 8 5 4 1 490 7 1 3 5 105 8 6 3 1 196 3 sum 12 7 2 2 5 280 8 7 2 1 28 7 3 1 5 105 8 8 1 1 1 4 1 1 4 1 7 1 4 4 175 4 1 2 3 6 7 2 3 4 889 8 sum 54912 4 2 1 3 6 7 3 2 4 889 4 1 3 2 6 7 4 1 4 175 9 1 1 9 1 4 2 2 2 17 7 1 5 3 105 9 1 2 8 36 4 3 1 2 6 7 2 4 3 889 9 2 1 8 36 4 1 4 1 1 7 3 3 3 1694 9 1 3 7 336 4 2 3 1 6 7 4 2 3 889 9 2 2 7 882 4 3 2 1 6 7 5 1 3 105 9 3 1 7 336 4 4 1 1 1 7 1 6 2 21 9 1 4 6 1176 7 2 5 2 280 9 2 3 6 5754 4 sum 56 7 3 4 2 889 9 3 2 6 5754 7 4 3 2 889 9 4 1 6 1176 5 1 1 5 1 7 5 2 2 280 9 1 5 5 1764 5 1 2 4 10 7 6 1 2 21 9 2 4 5 13941 5 2 1 4 10 7 1 7 1 1 9 3 3 5 26004 5 1 3 3 20 7 2 6 1 21 9 4 2 5 13941 5 2 2 3 55 7 3 5 1 105 9 5 1 5 1764 5 3 1 3 20 7 4 4 1 175 9 1 6 4 1176 5 1 4 2 10 7 5 3 1 105 9 2 5 4 13941 5 2 3 2 55 7 6 2 1 21 9 3 4 4 42015 5 3 2 2 55 7 7 1 1 1 9 4 3 4 42015 5 4 1 2 10 9 5 2 4 13941 5 1 5 1 1 7 sum 9152 9 6 1 4 1176 5 2 4 1 10 9 1 7 3 336 5 3 3 1 20 8 1 1 8 1 9 2 6 3 5754 5 4 2 1 10 8 1 2 7 28 9 3 5 3 26004 5 5 1 1 1 8 2 1 7 28 9 4 4 3 42015 8 1 3 6 196 9 5 3 3 26004 5 sum 288 8 2 2 6 518 9 6 2 3 5754 8 3 1 6 196 9 7 1 3 336 6 1 1 6 1 8 1 4 5 490 9 1 8 2 36 6 1 2 5 15 8 2 3 5 2436 9 2 7 2 882 6 2 1 5 15 8 3 2 5 2436 9 3 6 2 5754 6 1 3 4 50 8 4 1 5 490 9 4 5 2 13941 6 2 2 4 135 8 1 5 4 490 9 5 4 2 13941 6 3 1 4 50 8 2 4 4 3985 9 6 3 2 5754 6 1 4 3 50 8 3 3 4 7500 9 7 2 2 882 6 2 3 3 262 8 4 2 4 3985 9 8 1 2 36 6 3 2 3 262 8 5 1 4 490 9 1 9 1 1 6 4 1 3 50 8 1 6 3 196 9 2 8 1 36 1 1 1 1 1 1 1 m 0 9 9 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 m 1 0 216 Ars Math. Contemp. 15 (2018) 147-160 336 11 2 1 10 55 12 1 3 10 1210 1176 11 1 3 9 825 12 2 2 10 3135 1764 11 2 2 9 2145 12 3 1 10 1210 1176 11 3 1 9 825 12 1 4 9 9075 336 11 1 4 8 4950 12 2 3 9 43098 36 11 2 3 8 23694 12 3 2 9 43098 1 11 3 2 8 23694 12 4 1 9 9075 11 4 1 8 4950 12 1 5 8 32670 339456 11 1 5 7 13860 12 2 4 8 245223 11 2 4 7 105435 12 3 3 8 449988 1 11 3 3 7 194304 12 4 2 8 245223 45 11 4 2 7 105435 12 5 1 8 32670 45 11 5 1 7 13860 12 1 6 7 60984 540 11 1 6 6 19404 12 2 5 7 666996 1410 11 2 5 6 216601 12 3 4 7 1936308 540 11 3 4 6 634865 12 4 3 7 1936308 2520 11 4 3 6 634865 12 5 2 7 666996 12180 11 5 2 6 216601 12 6 1 7 60984 12180 11 6 1 6 19404 12 1 7 6 60984 2520 11 1 7 5 13860 12 2 6 6 925190 5292 11 2 6 5 216601 12 3 5 6 3915576 40935 11 3 5 5 931854 12 4 4 6 6195560 75840 11 4 4 5 1482250 12 5 3 6 3915576 40935 11 5 3 5 931854 12 6 2 6 925190 5292 11 6 2 5 216601 12 7 1 6 60984 5292 11 7 1 5 13860 12 1 8 5 32670 60626 11 1 8 4 4950 12 2 7 5 666996 179860 11 2 7 4 105435 12 3 6 5 3915576 179860 11 3 6 4 634865 12 4 5 5 9032898 60626 11 4 5 4 1482250 12 5 4 5 9032898 5292 11 5 4 4 1482250 12 6 3 5 3915576 2520 11 6 3 4 634865 12 7 2 5 666996 40935 11 7 2 4 105435 12 8 1 5 32670 179860 11 8 1 4 4950 12 1 9 4 9075 288025 11 1 9 3 825 12 2 8 4 245223 179860 11 2 8 3 23694 12 3 7 4 1936308 40935 11 3 7 3 194304 12 4 6 4 6195560 2520 11 4 6 3 634865 12 5 5 4 9032898 540 11 5 5 3 931854 12 6 4 4 6195560 12180 11 6 4 3 634865 12 7 3 4 1936308 75840 11 7 3 3 194304 12 8 2 4 245223 179860 11 8 2 3 23694 12 9 1 4 9075 179860 11 9 1 3 825 12 1 10 3 1210 75840 11 1 10 2 55 12 2 9 3 43098 12180 11 2 9 2 2145 12 3 8 3 449988 540 11 3 8 2 23694 12 4 7 3 1936308 45 11 4 7 2 105435 12 5 6 3 3915576 1410 11 5 6 2 216601 12 6 5 3 3915576 12180 11 6 5 2 216601 12 7 4 3 1936308 40935 11 7 4 2 105435 12 8 3 3 449988 60626 11 8 3 2 23694 12 9 2 3 43098 40935 11 9 2 2 2145 12 10 1 3 1210 12180 11 10 1 2 55 12 1 11 2 66 1410 11 1 11 1 1 12 2 10 2 3135 45 11 2 10 1 55 12 3 9 2 43098 1 11 3 9 1 825 12 4 8 2 245223 45 11 4 8 1 4950 12 5 7 2 666996 540 11 5 7 1 13860 12 6 6 2 925190 2520 11 6 6 1 19404 12 7 5 2 666996 5292 11 7 5 1 13860 12 8 4 2 245223 5292 11 8 4 1 4950 12 9 3 2 43098 2520 11 9 3 1 825 12 10 2 2 3135 540 11 10 2 1 55 12 11 1 2 66 45 11 11 1 1 1 12 1 12 1 1 1 12 2 11 1 66 11 sum 13891584 12 3 10 1 1210 2149888 12 4 9 1 9075 12 1 1 12 1 12 5 8 1 32670 1 12 1 2 11 66 12 6 7 1 60984 55 12 2 1 11 66 12 7 6 1 60984 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 253 12 8 5 1 32670 13 8 4 3 5264545 14 3 7 6 80231508 12 9 4 1 9075 13 9 3 3 960960 14 4 6 6 249321114 12 10 3 1 1210 13 10 2 3 74217 14 5 5 6 360078558 12 11 2 1 66 13 11 1 3 1716 14 6 4 6 249321114 12 12 1 1 1 13 1 12 2 78 14 7 3 6 80231508 13 2 11 2 4433 14 8 2 6 10701873 12 sum 91287552 13 3 10 2 74217 14 9 1 6 429429 13 4 9 2 525525 14 1 10 5 143143 13 1 1 13 1 13 5 8 2 1827683 14 2 9 5 4557553 13 1 2 12 78 13 6 7 2 3356522 14 3 8 5 44221632 13 2 1 12 78 13 7 6 2 3356522 14 4 7 5 181925268 13 1 3 11 1716 13 8 5 2 1827683 14 5 6 5 360078558 13 2 2 11 4433 13 9 4 2 525525 14 6 5 5 360078558 13 3 1 11 1716 13 10 3 2 74217 14 7 4 5 181925268 13 1 4 10 15730 13 11 2 2 4433 14 8 3 5 44221632 13 2 3 10 74217 13 12 1 2 78 14 9 2 5 4557553 13 3 2 10 74217 13 1 13 1 1 14 10 1 5 143143 13 4 1 10 15730 13 2 12 1 78 14 1 11 4 26026 13 1 5 9 70785 13 3 11 1 1716 14 2 10 4 1053052 13 2 4 9 525525 13 4 10 1 15730 14 3 9 4 13043030 13 3 3 9 960960 13 5 9 1 70785 14 4 8 4 69432090 13 4 2 9 525525 13 6 8 1 169884 14 5 7 4 181925268 13 5 1 9 70785 13 7 7 1 226512 14 6 6 4 249321114 13 1 6 8 169884 13 8 6 1 169884 14 7 5 4 181925268 13 2 5 8 1827683 13 9 5 1 70785 14 8 4 4 69432090 13 3 4 8 5264545 13 10 4 1 15730 14 9 3 4 13043030 13 4 3 8 5264545 13 11 3 1 1716 14 10 2 4 1053052 13 5 2 8 1827683 13 12 2 1 78 14 11 1 4 26026 13 6 1 8 169884 13 13 1 1 1 14 1 12 3 2366 13 1 7 7 226512 14 2 11 3 122122 13 2 6 7 3356522 13 sum 608583680 14 3 10 3 1919918 13 3 5 7 14019928 14 4 9 3 13043030 13 4 4 7 22089600 14 1 1 14 1 14 5 8 3 44221632 13 5 3 7 14019928 14 1 2 13 91 14 6 7 3 80231508 13 6 2 7 3356522 14 2 1 13 91 14 7 6 3 80231508 13 7 1 7 226512 14 1 3 12 2366 14 8 5 3 44221632 13 1 8 6 169884 14 2 2 12 6097 14 9 4 3 13043030 13 2 7 6 3356522 14 3 1 12 2366 14 10 3 3 1919918 13 3 6 6 19315114 14 1 4 11 26026 14 11 2 3 122122 13 4 5 6 44136820 14 2 3 11 122122 14 12 1 3 2366 13 5 4 6 44136820 14 3 2 11 122122 14 1 13 2 91 13 6 3 6 19315114 14 4 1 11 26026 14 2 12 2 6097 13 7 2 6 3356522 14 1 5 10 143143 14 3 11 2 122122 13 8 1 6 169884 14 2 4 10 1053052 14 4 10 2 1053052 13 1 9 5 70785 14 3 3 10 1919918 14 5 9 2 4557553 13 2 8 5 1827683 14 4 2 10 1053052 14 6 8 2 10701873 13 3 7 5 14019928 14 5 1 10 143143 14 7 7 2 14168988 13 4 6 5 44136820 14 1 6 9 429429 14 8 6 2 10701873 13 5 5 5 64013222 14 2 5 9 4557553 14 9 5 2 4557553 13 6 4 5 44136820 14 3 4 9 13043030 14 10 4 2 1053052 13 7 3 5 14019928 14 4 3 9 13043030 14 11 3 2 122122 13 8 2 5 1827683 14 5 2 9 4557553 14 12 2 2 6097 13 9 1 5 70785 14 6 1 9 429429 14 13 1 2 91 13 1 10 4 15730 14 1 7 8 736164 14 1 14 1 1 13 2 9 4 525525 14 2 6 8 10701873 14 2 13 1 91 13 3 8 4 5264545 14 3 5 8 44221632 14 3 12 1 2366 13 4 7 4 22089600 14 4 4 8 69432090 14 4 11 1 26026 13 5 6 4 44136820 14 5 3 8 44221632 14 5 10 1 143143 13 6 5 4 44136820 14 6 2 8 10701873 14 6 9 1 429429 13 7 4 4 22089600 14 7 1 8 736164 14 7 8 1 736164 13 8 3 4 5264545 14 1 8 7 736164 14 8 7 1 736164 13 9 2 4 525525 14 2 7 7 14168988 14 9 6 1 429429 13 10 1 4 15730 14 3 6 7 80231508 14 10 5 1 143143 13 1 11 3 1716 14 4 5 7 181925268 14 11 4 1 26026 13 2 10 3 74217 14 5 4 7 181925268 14 12 3 1 2366 13 3 9 3 960960 14 6 3 7 80231508 14 13 2 1 91 13 4 8 3 5264545 14 7 2 7 14168988 14 14 1 1 1 13 5 7 3 14019928 14 8 1 7 736164 13 6 6 3 19315114 14 1 9 6 429429 14 sum 4107939840 13 7 5 3 14019928 14 2 8 6 10701873 254 Ars Math. Contemp. 15 (2G1S) 225-266 A.2 Genus 1 d v e f h a 2 S l l470 l0 l a l 330 3 l l l l a 3 4 l 44l0 l0 2 7 l 6930 a 4 3 l 44l0 l0 3 6 l 4lSa0 3 sum l a S 2 l l470 l0 4 S l 97020 a 6 l l l26 l0 S 4 l 97020 4 l l 2 S l0 6 3 l 4lSa0 4 l 2 l S a sum l3l307 l0 7 2 l 6930 4 2 l l S 9 l l 7 2l0 l0 a l l 330 4 sum lS 9 9 l 2 2 l 6 6 3360 3360 l0 sum 97l3a3S S l l 3 lS 9 l 3 S l4700 ll l l 9 49S S l 2 2 40 9 2 2 S 3703S ll l 2 a l3200 S 2 l 2 40 9 3 l S l4700 ll 2 l a l3200 S l 3 l lS 9 l 4 4 23S20 ll l 3 7 l039S0 S 2 2 l 40 9 2 3 4 l0a2aS ll 2 2 7 2S90l7 S 3 l l lS 9 3 2 4 l0a2aS ll 3 l 7 l039S0 9 4 l 4 23S20 ll l 4 6 332640 S sum l6S 9 l S 3 l4700 ll 2 3 6 l493S2S 9 2 4 3 l0a2aS ll 3 2 6 l493S2S 6 l l 4 3S 9 3 3 3 l97a96 ll 4 l 6 332640 6 l 2 3 l7S 9 4 2 3 l0a2aS ll l S S 4aSl00 6 2 l 3 l7S 9 S l 3 l4700 ll 2 4 S 3420a3S 6 l 3 2 l7S 9 l 6 2 3360 ll 3 3 S 6l6S47a 6 2 2 2 4S6 9 2 S 2 3703S ll 4 2 S 3420a3S 6 3 l 2 l7S 9 3 4 2 l0a2aS ll S l S 4aSl00 6 l 4 l 3S 9 4 3 2 l0a2aS ll l 6 4 332640 6 2 3 l l7S 9 S 2 2 3703S ll 2 S 4 3420a3S 6 3 2 l l7S 9 6 l 2 3360 ll 3 4 4 96a4433 6 4 l l 3S 9 l 7 l 2l0 ll 4 3 4 96a4433 9 2 6 l 3360 ll S 2 4 3420a3S 6 sum l6ll 9 3 S l l4700 ll 6 l 4 332640 9 4 4 l 23S20 ll l 7 3 l039S0 7 l l S 70 9 S 3 l l4700 ll 2 6 3 l493S2S 7 l 2 4 S60 9 6 2 l 3360 ll 3 S 3 6l6S47a 7 2 l 4 S60 9 7 l l 2l0 ll 4 4 3 96a4433 7 l 3 3 l0S0 ll S 3 3 6l6S47a 7 2 2 3 269S 9 sum ll3a26l ll 6 2 3 l493S2S 7 3 l 3 l0S0 ll 7 l 3 l039S0 7 l 4 2 S60 l0 l l a 330 ll l a 2 l3200 7 2 3 2 269S l0 l 2 7 6930 ll 2 7 2 2S90l7 7 3 2 2 269S l0 2 l 7 6930 ll 3 6 2 l493S2S 7 4 l 2 S60 l0 l 3 6 4lSa0 ll 4 S 2 3420a3S 7 l S l 70 l0 2 2 6 l04llS ll S 4 2 3420a3S 7 2 4 l S60 l0 3 l 6 4lSa0 ll 6 3 2 l493S2S 7 3 3 l l0S0 l0 l 4 S 97020 ll 7 2 2 2S90l7 7 4 2 l S60 l0 2 3 S 440440 ll a l 2 l3200 7 S l l 70 l0 3 2 S 440440 ll l 9 l 49S l0 4 l S 97020 ll 2 a l l3200 7 sum l4a0S l0 l S 4 97020 ll 3 7 l l039S0 l0 2 4 4 6972S0 ll 4 6 l 332640 a l l 6 l26 l0 3 3 4 l2643l0 ll S S l 4aSl00 a l 2 S l470 l0 4 2 4 6972S0 ll 6 4 l 332640 a 2 l S l470 l0 S l 4 97020 ll 7 3 l l039S0 a l 3 4 44l0 l0 l 6 3 4lSa0 ll a 2 l l3200 a 2 2 4 lll99 l0 2 S 3 440440 ll 9 l l 49S a 3 l 4 44l0 l0 3 4 3 l2643l0 a l 4 3 44l0 l0 4 3 3 l2643l0 ll sum al96a469 a 2 3 3 206a4 l0 S 2 3 440440 a 3 2 3 206a4 l0 6 l 3 4lSa0 l2 l l l0 7lS a 4 l 3 44l0 l0 l 7 2 6930 l2 l 2 9 23S9S a l S 2 l470 l0 2 6 2 l04llS l2 2 l 9 23S9S a 2 4 2 lll99 l0 3 S 2 440440 l2 l 3 a 23S9S0 a 3 3 2 206a4 l0 4 4 2 6972S0 l2 2 2 a SaSSaS a 4 2 2 lll99 l0 S 3 2 440440 l2 3 l a 23S9S0 a S l 2 l470 l0 6 2 2 l04llS l2 l 4 7 990990 a l 6 l l26 l0 7 l 2 6930 l2 2 3 7 44l0l20 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 255 2 7 4410120 13 3 4 6 260619268 14 1 6 7 38648610 1 7 990990 13 4 3 6 260619268 14 2 5 7 375707570 5 6 1981980 13 5 2 6 93880696 14 3 4 7 1035514340 4 6 13768300 13 6 1 6 9513504 14 4 3 7 1035514340 3 6 24695580 13 1 7 5 6936930 14 5 2 7 375707570 2 6 13768300 13 2 6 5 93880696 14 6 1 7 38648610 1 6 1981980 13 3 5 5 374805834 14 1 7 6 38648610 6 5 1981980 13 4 4 5 582408775 14 2 6 6 512104880 5 5 19920390 13 5 3 5 374805834 14 3 5 6 2020140430 4 5 55785870 13 6 2 5 93880696 14 4 4 6 3126887407 3 5 55785870 13 7 1 5 6936930 14 5 3 6 2020140430 2 5 19920390 13 1 8 4 2642640 14 6 2 6 512104880 1 5 1981980 13 2 7 4 47604648 14 7 1 6 38648610 7 4 990990 13 3 6 4 260619268 14 1 8 5 21471450 6 4 13768300 13 4 5 4 582408775 14 2 7 5 375707570 5 4 55785870 13 5 4 4 582408775 14 3 6 5 2020140430 4 4 87100531 13 6 3 4 260619268 14 4 5 5 4475516612 3 4 55785870 13 7 2 4 47604648 14 5 4 5 4475516612 2 4 13768300 13 8 1 4 2642640 14 6 3 5 2020140430 1 4 990990 13 1 9 3 495495 14 7 2 5 375707570 8 3 235950 13 2 8 3 11674663 14 8 1 5 21471450 7 3 4410120 13 3 7 3 85050784 14 1 9 4 6441435 6 3 24695580 13 4 6 3 260619268 14 2 8 4 145864355 5 3 55785870 13 5 5 3 374805834 14 3 7 4 1035514340 4 3 55785870 13 6 4 3 260619268 14 4 6 4 3126887407 3 3 24695580 13 7 3 3 85050784 14 5 5 4 4475516612 2 3 4410120 13 8 2 3 11674663 14 6 4 4 3126887407 1 3 235950 13 9 1 3 495495 14 7 3 4 1035514340 9 2 23595 13 1 10 2 40040 14 8 2 4 145864355 8 2 585585 13 2 9 2 1225653 14 9 1 4 6441435 7 2 4410120 13 3 8 2 11674663 14 1 10 3 975975 6 2 13768300 13 4 7 2 47604648 14 2 9 3 28283255 5 2 19920390 13 5 6 2 93880696 14 3 8 3 259750218 4 2 13768300 13 6 5 2 93880696 14 4 7 3 1035514340 3 2 4410120 13 7 4 2 47604648 14 5 6 3 2020140430 2 2 585585 13 8 3 2 11674663 14 6 5 3 2020140430 1 2 23595 13 9 2 2 1225653 14 7 4 3 1035514340 10 1 715 13 10 1 2 40040 14 8 3 3 259750218 9 1 23595 13 1 11 1 1001 14 9 2 3 28283255 8 1 235950 13 2 10 1 40040 14 10 1 3 975975 7 1 990990 13 3 9 1 495495 14 1 11 2 65065 6 1 1981980 13 4 8 1 2642640 14 2 10 2 2407405 5 1 1981980 13 5 7 1 6936930 14 3 9 2 28283255 4 1 990990 13 6 6 1 9513504 14 4 8 2 145864355 3 1 235950 13 7 5 1 6936930 14 5 7 2 375707570 2 1 23595 13 8 4 1 2642640 14 6 6 2 512104880 1 1 715 13 9 3 1 495495 14 7 5 2 375707570 13 10 2 1 40040 14 8 4 2 145864355 sum 685888171 13 11 1 1 1001 14 9 3 2 28283255 14 10 2 2 2407405 1 11 1001 13 sum 5702382933 14 11 1 2 65065 2 10 40040 14 1 12 1 1365 1 10 40040 14 1 1 12 1365 14 2 11 1 65065 3 9 495495 14 1 2 11 65065 14 3 10 1 975975 2 9 1225653 14 2 1 11 65065 14 4 9 1 6441435 1 9 495495 14 1 3 10 975975 14 5 8 1 21471450 4 8 2642640 14 2 2 10 2407405 14 6 7 1 38648610 3 8 11674663 14 3 1 10 975975 14 7 6 1 38648610 2 8 11674663 14 1 4 9 6441435 14 8 5 1 21471450 1 8 2642640 14 2 3 9 28283255 14 9 4 1 6441435 5 7 6936930 14 3 2 9 28283255 14 10 3 1 975975 4 7 47604648 14 4 1 9 6441435 14 11 2 1 65065 3 7 85050784 14 1 5 8 21471450 14 12 1 1 1365 2 7 47604648 14 2 4 8 145864355 1 7 6936930 14 3 3 8 259750218 14 sum 4716867857 6 6 9513504 14 4 2 8 145864355 5 6 93880696 14 5 1 8 21471450 256 Ars Math. Contemp. 15 (2018) 147-160 A.3 Genus 2 d v e f h l0 2 S l l670l3 l2 l a l aaa03 S l l l a l0 3 4 l 47l240 l2 2 7 l lSaSSa4 l0 4 3 l 47l240 l2 3 6 l 8 6S4 64 6 S sum a l0 S 2 l l670l3 l2 4 S l l932430S l0 6 l l l640l l2 S 4 l l932430S 6 l l 2 a4 l2 6 3 l 8 6S4 64 6 6 l 2 l a4 l0 sum l3S4S2l6 l2 7 2 l lSaSSa4 6 2 l l a4 ll l l 7 39963 l2 a l l aaa03 6 sum 2S2 ll ll l 2 2 l 6 6 SS00ll SS00ll l2 sum la0S0l094a 7 l l 3 469 ll l 3 S 222l06S l3 l l 9 la3la3 7 l 2 2 1183 ll 2 2 S S4090l9 l3 l 2 a 4ll4ll0 7 2 l 2 1183 ll 3 l S 222l06S l3 2 l a 4ll4ll0 7 l 3 l 469 ll l 4 4 346S000 l3 l 3 7 29l3Sl06 7 2 2 l 1183 ll 2 3 4 lS0l4a46 l3 2 2 7 70367479 7 3 l l 469 ll 3 2 4 lS0l4a46 l3 3 l 7 29l3Sl06 ll 4 l 4 346S000 l3 l 4 6 a7933a46 7 sum 4 9S6 ll l S 3 222l06S l3 2 3 6 374l27663 ll 2 4 3 lS0l4a46 l3 3 2 6 374l27663 a l l 4 la69 ll 3 3 3 267l74a2 l3 4 l 6 a7933a46 a l 2 3 aS26 ll 4 2 3 lS0l4a46 l3 l S S l2SaSS730 a 2 l 3 aS26 ll S l 3 222l06S l3 2 4 S a24962S02 a l 3 2 aS26 ll l 6 2 SS00ll l3 3 3 S l4S34l4a46 a 2 2 2 2l229 ll 2 S 2 S4090l9 l3 4 2 S a24962S02 a 3 l 2 aS26 ll 3 4 2 lS0l4a46 l3 S l S l2SaSS730 a l 4 l la69 ll 4 3 2 lS0l4a46 l3 l 6 4 a7933a46 a 2 3 l aS26 ll S 2 2 S4090l9 l3 2 S 4 a24962S02 a 3 2 l aS26 ll 6 l 2 SS00ll l3 3 4 4 22392a0420 a 4 l l la69 ll l 7 l 39963 l3 4 3 4 22392a0420 ll 2 6 l SS00ll l3 S 2 4 a24962S02 a sum 77992 ll 3 S l 222l06S l3 6 l 4 a7933a46 ll 4 4 l 346S000 l3 l 7 3 29l3Sl06 9 l l S S9aS ll S 3 l 222l06S l3 2 6 3 374l27663 9 l 2 4 42Saa ll 6 2 l SS00ll l3 3 S 3 l4S34l4a46 9 2 l 4 42Saa ll 7 l l 39963 l3 4 4 3 22392a0420 9 l 3 3 7702a l3 S 3 3 l4S34l4a46 9 2 2 3 la9999 ll sum l60l74960 l3 6 2 3 374l27663 9 3 l 3 7702a l3 7 l 3 29l3Sl06 9 l 4 2 42Saa l2 l l a aaa03 l3 l a 2 4ll4ll0 9 2 3 2 la9999 l2 l 2 7 lSaSSa4 l3 2 7 2 70367479 9 3 2 2 la9999 l2 2 l 7 lSaSSa4 l3 3 6 2 374l27663 9 4 l 2 42Saa l2 l 3 6 a6S4646 l3 4 S 2 a24962S02 9 l S l S9aS l2 2 2 6 209al337 l3 S 4 2 a24962S02 9 2 4 l 42Saa l2 3 l 6 a6S4646 l3 6 3 2 374l27663 9 3 3 l 7702a l2 l 4 S l932430S l3 7 2 2 70367479 9 4 2 l 42Saa l2 2 3 S a2a97296 l3 a l 2 4ll4ll0 9 S l l S9aS l2 3 2 S a2a97296 l3 l 9 l la3la3 l2 4 l S l932430S l3 2 a l 4ll4ll0 9 sum l074S64 l2 l S 4 l932430S l3 3 7 l 29l3Sl06 l2 2 4 4 l2a420004 l3 4 6 l a7933a46 l0 l l 6 l640l l2 3 3 4 2272S6Sl0 l3 S S l l2SaSS730 l0 l 2 S 167013 l2 4 2 4 l2a420004 l3 6 4 l a7933a46 l0 2 l S 167013 l2 S l 4 l932430S l3 7 3 l 29l3Sl06 l0 l 3 4 47l240 l2 l 6 3 a6S4646 l3 a 2 l 4ll4ll0 l0 2 2 4 llS409S l2 2 S 3 a2a97296 l3 9 l l la3la3 l0 3 l 4 47l240 l2 3 4 3 2272S6Sl0 l0 l 4 3 47l240 l2 4 3 3 2272S6Sl0 l3 sum l9Saa944336 l0 2 3 3 206a070 l2 S 2 3 a2a97296 l0 3 2 3 206a070 l2 6 l 3 a6S4646 l4 l l l0 3SS3SS l0 4 l 3 47l240 l2 l 7 2 lSaSSa4 l4 l 2 9 979a7a9 l0 l S 2 167013 l2 2 6 2 209al337 l4 2 l 9 979a7a9 l0 2 4 2 llS409S l2 3 S 2 a2a97296 l4 l 3 a a729l204 l0 3 3 2 206a070 l2 4 4 2 l2a420004 l4 2 2 a 2l0l64227 l0 4 2 2 llS409S l2 S 3 2 a2a97296 l4 3 l a a729l204 l0 S l 2 167013 l2 6 2 2 209al337 l4 l 4 7 34la2S4a4 l0 l 6 l l640l l2 7 l 2 lSaSSa4 l4 2 3 7 l4444326l2 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 257 14 3 2 7 1444432612 14 5 3 4 16427471172 14 7 3 2 1444432612 14 4 1 7 341825484 14 6 2 4 4286172247 14 8 2 2 210164227 14 1 5 6 661320660 14 7 1 4 341825484 14 9 1 2 9798789 14 2 4 6 4286172247 14 1 8 3 87291204 14 1 10 1 355355 14 3 3 6 7523770016 14 2 7 3 1444432612 14 2 9 1 9798789 14 4 2 6 4286172247 14 3 6 3 7523770016 14 3 8 1 87291204 14 5 1 6 661320660 14 4 5 3 16427471172 14 4 7 1 341825484 14 1 6 5 661320660 14 5 4 3 16427471172 14 5 6 1 661320660 14 2 5 5 6100939726 14 6 3 3 7523770016 14 6 5 1 661320660 14 3 4 5 16427471172 14 7 2 3 1444432612 14 7 4 1 341825484 14 4 3 5 16427471172 14 8 1 3 87291204 14 8 3 1 87291204 14 5 2 5 6100939726 14 1 9 2 9798789 14 9 2 1 9798789 14 6 1 5 661320660 14 2 8 2 210164227 14 10 1 1 355355 14 1 7 4 341825484 14 3 7 2 1444432612 14 2 6 4 4286172247 14 4 6 2 4286172247 14 sum 206254571236 14 3 5 4 16427471172 14 5 5 2 6100939726 14 4 4 4 25199010256 14 6 4 2 4286172247 A.4 Genus 3 d v e f h 13 1 7 1 8691683 7 1 1 1 180 11 sum 112868844 13 2 6 1 108452916 13 3 5 1 414918075 7 sum 180 12 1 1 6 2641925 13 4 4 1 636184120 12 1 2 5 24656775 13 5 3 1 414918075 8 1 1 2 3044 12 2 1 5 24656775 13 6 2 1 108452916 8 1 2 1 3044 12 1 3 4 66805310 13 7 1 1 8691683 8 2 1 1 3044 12 2 2 4 159762815 12 3 1 4 66805310 13 sum 28540603884 8 sum 9132 12 1 4 3 66805310 12 2 3 3 280514670 14 1 1 8 25537655 9 1 1 3 26060 12 3 2 3 280514670 14 1 2 7 409732895 9 1 2 2 63600 12 4 1 3 66805310 14 2 1 7 409732895 9 2 1 2 63600 12 1 5 2 24656775 14 1 3 6 2096068975 9 1 3 1 26060 12 2 4 2 159762815 14 2 2 6 4973691275 9 2 2 1 63600 12 3 3 2 280514670 14 3 1 6 2096068975 9 3 1 1 26060 12 4 2 2 159762815 14 1 4 5 4538348815 12 5 1 2 24656775 14 2 3 5 18733893115 9 sum 268980 12 1 6 1 2641925 14 3 2 5 18733893115 12 2 5 1 24656775 14 4 1 5 4538348815 10 1 1 4 152900 12 3 4 1 66805310 14 1 5 4 4538348815 10 1 2 3 659340 12 4 3 1 66805310 14 2 4 4 28579309570 10 2 1 3 659340 12 5 2 1 24656775 14 3 3 4 49719495672 10 1 3 2 659340 12 6 1 1 2641925 14 4 2 4 28579309570 10 2 2 2 1595480 14 5 1 4 4538348815 10 3 1 2 659340 12 sum 1877530740 14 1 6 3 2096068975 10 1 4 1 152900 14 2 5 3 18733893115 10 2 3 1 659340 13 1 1 7 8691683 14 3 4 3 49719495672 10 3 2 1 659340 13 1 2 6 108452916 14 4 3 3 49719495672 10 4 1 1 152900 13 2 1 6 108452916 14 5 2 3 18733893115 13 1 3 5 414918075 14 6 1 3 2096068975 10 sum 6010220 13 2 2 5 988043771 14 1 7 2 409732895 13 3 1 5 414918075 14 2 6 2 4973691275 11 1 1 5 696905 13 1 4 4 636184120 14 3 5 2 18733893115 11 1 2 4 4606910 13 2 3 4 2646424729 14 4 4 2 28579309570 11 2 1 4 4606910 13 3 2 4 2646424729 14 5 3 2 18733893115 11 1 3 3 8141100 13 4 1 4 636184120 14 6 2 2 4973691275 11 2 2 3 19571123 13 1 5 3 414918075 14 7 1 2 409732895 11 3 1 3 8141100 13 2 4 3 2646424729 14 1 8 1 25537655 11 1 4 2 4606910 13 3 3 3 4623070842 14 2 7 1 409732895 11 2 3 2 19571123 13 4 2 3 2646424729 14 3 6 1 2096068975 11 3 2 2 19571123 13 5 1 3 414918075 14 4 5 1 4538348815 11 4 1 2 4606910 13 1 6 2 108452916 14 5 4 1 4538348815 11 1 5 1 696905 13 2 5 2 988043771 14 6 3 1 2096068975 11 2 4 1 4606910 13 3 4 2 2646424729 14 7 2 1 409732895 11 3 3 1 8141100 13 4 3 2 2646424729 14 8 1 1 25537655 11 4 2 1 4606910 13 5 2 2 988043771 11 5 1 1 696905 13 6 1 2 108452916 14 sum 404562365316 258 Ars Math. Contemp. 15 (2018) 147-160 A.5 Genus 4 d v e f h 12 3 1 2 75220860 9 1 1 1 8064 12 1 4 1 18128396 14 1 1 6 539651112 12 2 3 1 75220860 14 1 2 5 4736419688 9 sum 8064 12 3 2 1 75220860 14 2 1 5 4736419688 12 4 1 1 18128396 14 1 3 4 12465308856 10 1 1 2 193248 14 2 2 4 29310854804 10 1 2 1 193248 12 sum 684173164 14 3 1 4 12465308856 10 2 1 1 193248 14 1 4 3 12465308856 13 1 1 5 109425316 14 2 3 3 50713072144 10 sum 579744 13 1 2 4 687238552 14 3 2 3 50713072144 13 2 1 4 687238552 14 4 1 3 12465308856 11 1 1 3 2286636 13 1 3 3 1194737544 14 1 5 2 4736419688 11 1 2 2 5458464 13 2 2 3 2820651496 14 2 4 2 29310854804 11 2 1 2 5458464 13 3 1 3 1194737544 14 3 3 2 50713072144 11 1 3 1 2286636 13 1 4 2 687238552 14 4 2 2 29310854804 11 2 2 1 5458464 13 2 3 2 2820651496 14 5 1 2 4736419688 11 3 1 1 2286636 13 3 2 2 2820651496 14 1 6 1 539651112 13 4 1 2 687238552 14 2 5 1 4736419688 11 sum 23235300 13 1 5 1 109425316 14 3 4 1 12465308856 13 2 4 1 687238552 14 4 3 1 12465308856 12 1 1 4 18128396 13 3 3 1 1194737544 14 5 2 1 4736419688 12 1 2 3 75220860 13 4 2 1 687238552 14 6 1 1 539651112 12 2 1 3 75220860 13 5 1 1 109425316 12 1 3 2 75220860 14 sum 344901105444 12 2 2 2 178462816 13 sum 16497874380 A.6 Genus 5 d v e f h 13 1 1 3 292271616 14 2 1 3 11947069680 11 1 1 1 604800 13 1 2 2 686597184 14 1 3 2 11947069680 13 2 1 2 686597184 14 2 2 2 27934773440 11 sum 604800 13 1 3 1 292271616 14 3 1 2 11947069680 13 2 2 1 686597184 14 1 4 1 2961802480 12 1 1 2 19056960 13 3 1 1 292271616 14 2 3 1 11947069680 12 1 2 1 19056960 14 3 2 1 11947069680 12 2 1 1 19056960 13 sum 2936606400 14 4 1 1 2961802480 12 sum 57170880 14 1 1 4 2961802480 14 sum 108502598960 14 1 2 3 11947069680 A.7 Genus 6 d v e f h 13 1 1 1 68428800 14 1 1 2 2699672832 14 sum 8099018496 14 1 2 1 2699672832 13 sum 68428800 14 2 1 1 2699672832 These tables extend to 14 darts the part of Appendix B of [24] about rooted hypermaps. A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 259 B First numbers of unrooted hypermaps The following sections show the numbers H of unrooted hypermaps of genus g with d darts, v vertices, e edges and d - v - e + 2(1 - g) faces, for g < 6 and d < 14. B.1 Genus 0 d v e f H 6 1 5 2 3 8 2 5 3 309 1 1 1 1 1 6 2 4 2 24 8 3 4 3 946 6 3 3 2 46 8 4 3 3 946 1 sum 1 6 4 2 2 24 8 5 2 3 309 6 5 1 2 3 8 6 1 3 26 2 1 1 2 1 6 1 6 1 1 8 1 7 2 4 2 1 2 1 1 6 2 5 1 3 8 2 6 2 67 2 2 1 1 1 6 3 4 1 10 8 3 5 2 309 6 4 3 1 10 8 4 4 2 505 2 sum 3 6 5 2 1 3 8 5 3 2 309 6 6 1 1 1 8 6 2 2 67 3 1 1 3 1 8 7 1 2 4 3 1 2 2 1 6 sum 291 8 1 8 1 1 3 2 1 2 1 8 2 7 1 4 3 1 3 1 1 7 1 1 7 1 8 3 6 1 26 3 2 2 1 1 7 1 2 6 3 8 4 5 1 64 3 3 1 1 1 7 2 1 6 3 8 5 4 1 64 7 1 3 5 15 8 6 3 1 26 3 sum 6 7 2 2 5 40 8 7 2 1 4 7 3 1 5 15 8 8 1 1 1 4 1 1 4 1 7 1 4 4 25 4 1 2 3 2 7 2 3 4 127 8 sum 6975 4 2 1 3 2 7 3 2 4 127 4 1 3 2 2 7 4 1 4 25 9 1 1 9 1 4 2 2 2 5 7 1 5 3 15 9 1 2 8 4 4 3 1 2 2 7 2 4 3 127 9 2 1 8 4 4 1 4 1 1 7 3 3 3 242 9 1 3 7 38 4 2 3 1 2 7 4 2 3 127 9 2 2 7 98 4 3 2 1 2 7 5 1 3 15 9 3 1 7 38 4 4 1 1 1 7 1 6 2 3 9 1 4 6 132 7 2 5 2 40 9 2 3 6 640 4 sum 20 7 3 4 2 127 9 3 2 6 640 7 4 3 2 127 9 4 1 6 132 5 1 1 5 1 7 5 2 2 40 9 1 5 5 196 5 1 2 4 2 7 6 1 2 3 9 2 4 5 1549 5 2 1 4 2 7 1 7 1 1 9 3 3 5 2890 5 1 3 3 4 7 2 6 1 3 9 4 2 5 1549 5 2 2 3 11 7 3 5 1 15 9 5 1 5 196 5 3 1 3 4 7 4 4 1 25 9 1 6 4 132 5 1 4 2 2 7 5 3 1 15 9 2 5 4 1549 5 2 3 2 11 7 6 2 1 3 9 3 4 4 4671 5 3 2 2 11 7 7 1 1 1 9 4 3 4 4671 5 4 1 2 2 9 5 2 4 1549 5 1 5 1 1 7 sum 1310 9 6 1 4 132 5 2 4 1 2 9 1 7 3 38 5 3 3 1 4 8 1 1 8 1 9 2 6 3 640 5 4 2 1 2 8 1 2 7 4 9 3 5 3 2890 5 5 1 1 1 8 2 1 7 4 9 4 4 3 4671 8 1 3 6 26 9 5 3 3 2890 5 sum 60 8 2 2 6 67 9 6 2 3 640 8 3 1 6 26 9 7 1 3 38 6 1 1 6 1 8 1 4 5 64 9 1 8 2 4 6 1 2 5 3 8 2 3 5 309 9 2 7 2 98 6 2 1 5 3 8 3 2 5 309 9 3 6 2 640 6 1 3 4 10 8 4 1 5 64 9 4 5 2 1549 6 2 2 4 24 8 1 5 4 64 9 5 4 2 1549 6 3 1 4 10 8 2 4 4 505 9 6 3 2 640 6 1 4 3 10 8 3 3 4 946 9 7 2 2 98 6 2 3 3 46 8 4 2 4 505 9 8 1 2 4 6 3 2 3 46 8 5 1 4 64 9 1 9 1 1 6 4 1 3 10 8 1 6 3 26 9 2 8 1 4 1 1 1 1 1 1 1 m 0 9 9 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 m 1 0 224 Ars Math. Contemp. 15 (2018) 147-160 38 11 2 1 10 5 12 1 3 10 104 132 11 1 3 9 75 12 2 2 10 265 196 11 2 2 9 195 12 3 1 10 104 132 11 3 1 9 75 12 1 4 9 765 38 11 1 4 8 450 12 2 3 9 3605 4 11 2 3 8 2154 12 3 2 9 3605 1 11 3 2 8 2154 12 4 1 9 765 11 4 1 8 450 12 1 5 8 2736 37746 11 1 5 7 1260 12 2 4 8 20472 11 2 4 7 9585 12 3 3 8 37545 1 11 3 3 7 17664 12 4 2 8 20472 5 11 4 2 7 9585 12 5 1 8 2736 5 11 5 1 7 1260 12 1 6 7 5102 56 11 1 6 6 1764 12 2 5 7 55633 144 11 2 5 6 19691 12 3 4 7 161455 56 11 3 4 6 57715 12 4 3 7 161455 256 11 4 3 6 57715 12 5 2 7 55633 1226 11 5 2 6 19691 12 6 1 7 5102 1226 11 6 1 6 1764 12 1 7 6 5102 256 11 1 7 5 1260 12 2 6 6 77174 536 11 2 6 5 19691 12 3 5 6 326432 4111 11 3 5 5 84714 12 4 4 6 516507 7606 11 4 4 5 134750 12 5 3 6 326432 4111 11 5 3 5 84714 12 6 2 6 77174 536 11 6 2 5 19691 12 7 1 6 5102 536 11 7 1 5 1260 12 1 8 5 2736 6081 11 1 8 4 450 12 2 7 5 55633 18019 11 2 7 4 9585 12 3 6 5 326432 18019 11 3 6 4 57715 12 4 5 5 752940 6081 11 4 5 4 134750 12 5 4 5 752940 536 11 5 4 4 134750 12 6 3 5 326432 256 11 6 3 4 57715 12 7 2 5 55633 4111 11 7 2 4 9585 12 8 1 5 2736 18019 11 8 1 4 450 12 1 9 4 765 28852 11 1 9 3 75 12 2 8 4 20472 18019 11 2 8 3 2154 12 3 7 4 161455 4111 11 3 7 3 17664 12 4 6 4 516507 256 11 4 6 3 57715 12 5 5 4 752940 56 11 5 5 3 84714 12 6 4 4 516507 1226 11 6 4 3 57715 12 7 3 4 161455 7606 11 7 3 3 17664 12 8 2 4 20472 18019 11 8 2 3 2154 12 9 1 4 765 18019 11 9 1 3 75 12 1 10 3 104 7606 11 1 10 2 5 12 2 9 3 3605 1226 11 2 9 2 195 12 3 8 3 37545 56 11 3 8 2 2154 12 4 7 3 161455 5 11 4 7 2 9585 12 5 6 3 326432 144 11 5 6 2 19691 12 6 5 3 326432 1226 11 6 5 2 19691 12 7 4 3 161455 4111 11 7 4 2 9585 12 8 3 3 37545 6081 11 8 3 2 2154 12 9 2 3 3605 4111 11 9 2 2 195 12 10 1 3 104 1226 11 10 1 2 5 12 1 11 2 6 144 11 1 11 1 1 12 2 10 2 265 5 11 2 10 1 5 12 3 9 2 3605 1 11 3 9 1 75 12 4 8 2 20472 5 11 4 8 1 450 12 5 7 2 55633 56 11 5 7 1 1260 12 6 6 2 77174 256 11 6 6 1 1764 12 7 5 2 55633 536 11 7 5 1 1260 12 8 4 2 20472 536 11 8 4 1 450 12 9 3 2 3605 256 11 9 3 1 75 12 10 2 2 265 56 11 10 2 1 5 12 11 1 2 6 5 11 11 1 1 1 12 1 12 1 1 1 12 2 11 1 6 11 sum 1262874 12 3 10 1 104 215602 12 4 9 1 765 12 1 1 12 1 12 5 8 1 2736 1 12 1 2 11 6 12 6 7 1 5102 5 12 2 1 11 6 12 7 6 1 5102 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 261 12 8 5 1 2736 13 8 4 3 404965 14 3 7 6 5731330 12 9 4 1 765 13 9 3 3 73920 14 4 6 6 17809776 12 10 3 1 104 13 10 2 3 5709 14 5 5 6 25720986 12 11 2 1 6 13 11 1 3 132 14 6 4 6 17809776 12 12 1 1 1 13 1 12 2 6 14 7 3 6 5731330 13 2 11 2 341 14 8 2 6 764633 12 sum 7611156 13 3 10 2 5709 14 9 1 6 30711 13 4 9 2 40425 14 1 10 5 10247 13 1 1 13 1 13 5 8 2 140591 14 2 9 5 325652 13 1 2 12 6 13 6 7 2 258194 14 3 8 5 3159069 13 2 1 12 6 13 7 6 2 258194 14 4 7 5 12995424 13 1 3 11 132 13 8 5 2 140591 14 5 6 5 25720986 13 2 2 11 341 13 9 4 2 40425 14 6 5 5 25720986 13 3 1 11 132 13 10 3 2 5709 14 7 4 5 12995424 13 1 4 10 1210 13 11 2 2 341 14 8 3 5 3159069 13 2 3 10 5709 13 12 1 2 6 14 9 2 5 325652 13 3 2 10 5709 13 1 13 1 1 14 10 1 5 10247 13 4 1 10 1210 13 2 12 1 6 14 1 11 4 1868 13 1 5 9 5445 13 3 11 1 132 14 2 10 4 75283 13 2 4 9 40425 13 4 10 1 1210 14 3 9 4 931845 13 3 3 9 73920 13 5 9 1 5445 14 4 8 4 4960016 13 4 2 9 40425 13 6 8 1 13068 14 5 7 4 12995424 13 5 1 9 5445 13 7 7 1 17424 14 6 6 4 17809776 13 1 6 8 13068 13 8 6 1 13068 14 7 5 4 12995424 13 2 5 8 140591 13 9 5 1 5445 14 8 4 4 4960016 13 3 4 8 404965 13 10 4 1 1210 14 9 3 4 931845 13 4 3 8 404965 13 11 3 1 132 14 10 2 4 75283 13 5 2 8 140591 13 12 2 1 6 14 11 1 4 1868 13 6 1 8 13068 13 13 1 1 1 14 1 12 3 172 13 1 7 7 17424 14 2 11 3 8741 13 2 6 7 258194 13 sum 46814132 14 3 10 3 137217 13 3 5 7 1078456 14 4 9 3 931845 13 4 4 7 1699200 14 1 1 14 1 14 5 8 3 3159069 13 5 3 7 1078456 14 1 2 13 7 14 6 7 3 5731330 13 6 2 7 258194 14 2 1 13 7 14 7 6 3 5731330 13 7 1 7 17424 14 1 3 12 172 14 8 5 3 3159069 13 1 8 6 13068 14 2 2 12 440 14 9 4 3 931845 13 2 7 6 258194 14 3 1 12 172 14 10 3 3 137217 13 3 6 6 1485778 14 1 4 11 1868 14 11 2 3 8741 13 4 5 6 3395140 14 2 3 11 8741 14 12 1 3 172 13 5 4 6 3395140 14 3 2 11 8741 14 1 13 2 7 13 6 3 6 1485778 14 4 1 11 1868 14 2 12 2 440 13 7 2 6 258194 14 1 5 10 10247 14 3 11 2 8741 13 8 1 6 13068 14 2 4 10 75283 14 4 10 2 75283 13 1 9 5 5445 14 3 3 10 137217 14 5 9 2 325652 13 2 8 5 140591 14 4 2 10 75283 14 6 8 2 764633 13 3 7 5 1078456 14 5 1 10 10247 14 7 7 2 1012271 13 4 6 5 3395140 14 1 6 9 30711 14 8 6 2 764633 13 5 5 5 4924094 14 2 5 9 325652 14 9 5 2 325652 13 6 4 5 3395140 14 3 4 9 931845 14 10 4 2 75283 13 7 3 5 1078456 14 4 3 9 931845 14 11 3 2 8741 13 8 2 5 140591 14 5 2 9 325652 14 12 2 2 440 13 9 1 5 5445 14 6 1 9 30711 14 13 1 2 7 13 1 10 4 1210 14 1 7 8 52634 14 1 14 1 1 13 2 9 4 40425 14 2 6 8 764633 14 2 13 1 7 13 3 8 4 404965 14 3 5 8 3159069 14 3 12 1 172 13 4 7 4 1699200 14 4 4 8 4960016 14 4 11 1 1868 13 5 6 4 3395140 14 5 3 8 3159069 14 5 10 1 10247 13 6 5 4 3395140 14 6 2 8 764633 14 6 9 1 30711 13 7 4 4 1699200 14 7 1 8 52634 14 7 8 1 52634 13 8 3 4 404965 14 1 8 7 52634 14 8 7 1 52634 13 9 2 4 40425 14 2 7 7 1012271 14 9 6 1 30711 13 10 1 4 1210 14 3 6 7 5731330 14 10 5 1 10247 13 1 11 3 132 14 4 5 7 12995424 14 11 4 1 1868 13 2 10 3 5709 14 5 4 7 12995424 14 12 3 1 172 13 3 9 3 73920 14 6 3 7 5731330 14 13 2 1 7 13 4 8 3 404965 14 7 2 7 1012271 14 14 1 1 1 13 5 7 3 1078456 14 8 1 7 52634 13 6 6 3 1485778 14 1 9 6 30711 14 sum 293447817 13 7 5 3 1078456 14 2 8 6 764633 l f 1 m 2 1 1 m 3 2 2 1 1 1 m 4 3 3 2 2 2 1 1 1 1 m 5 4 4 3 3 3 2 2 2 2 1 1 1 1 1 m 6 5 5 4 4 4 3 3 3 3 2 2 2 2 2 1 Ars Math. Contemp. 15 (2018) 225-266 7 31 31 31 78 31 7 31 31 7 285 10 80 80 150 385 150 80 385 385 80 10 80 150 80 10 2115 17 187 187 557 1409 557 557 2597 2597 557 187 1409 2597 1409 187 17 8 2 5 1 187 10 1 8 1 34 8 3 4 1 557 10 2 7 1 698 8 4 3 1 557 10 3 6 1 4172 8 5 2 1 187 10 4 5 1 9724 8 6 1 1 17 10 5 4 1 9724 10 6 3 1 4172 8 sum 16533 10 7 2 1 698 10 8 1 1 34 9 1 1 7 24 9 1 2 6 374 10 sum 972441 9 2 1 6 374 9 1 3 5 1634 11 1 1 9 45 9 2 2 5 4115 11 1 2 8 1200 9 3 1 5 1634 11 2 1 8 1200 9 1 4 4 2616 11 1 3 7 9450 9 2 3 4 12033 11 2 2 7 23547 9 3 2 4 12033 11 3 1 7 9450 9 4 1 4 2616 11 1 4 6 30240 9 1 5 3 1634 11 2 3 6 135775 9 2 4 3 12033 11 3 2 6 135775 9 3 3 3 21990 11 4 1 6 30240 9 4 2 3 12033 11 1 5 5 44100 9 5 1 3 1634 11 2 4 5 310985 9 1 6 2 374 11 3 3 5 560498 9 2 5 2 4115 11 4 2 5 310985 9 3 4 2 12033 11 5 1 5 44100 9 4 3 2 12033 11 1 6 4 30240 9 5 2 2 4115 11 2 5 4 310985 9 6 1 2 374 11 3 4 4 880403 9 1 7 1 24 11 4 3 4 880403 9 2 6 1 374 11 5 2 4 310985 9 3 5 1 1634 11 6 1 4 30240 9 4 4 1 2616 11 1 7 3 9450 9 5 3 1 1634 11 2 6 3 135775 9 6 2 1 374 11 3 5 3 560498 9 7 1 1 24 11 4 4 3 880403 11 5 3 3 560498 9 sum 126501 11 6 2 3 135775 11 7 1 3 9450 10 1 1 8 34 11 1 8 2 1200 10 1 2 7 698 11 2 7 2 23547 10 2 1 7 698 11 3 6 2 135775 10 1 3 6 4172 11 4 5 2 310985 10 2 2 6 10434 11 5 4 2 310985 10 3 1 6 4172 11 6 3 2 135775 10 1 4 5 9724 11 7 2 2 23547 10 2 3 5 44091 11 8 1 2 1200 10 3 2 5 44091 11 1 9 1 45 10 4 1 5 9724 11 2 8 1 1200 10 1 5 4 9724 11 3 7 1 9450 10 2 4 4 69790 11 4 6 1 30240 10 3 3 4 126519 11 5 5 1 44100 10 4 2 4 69790 11 6 4 1 30240 10 5 1 4 9724 11 7 3 1 9450 10 1 6 3 4172 11 8 2 1 1200 10 2 5 3 44091 11 9 1 1 45 10 3 4 3 126519 10 4 3 3 126519 11 sum 7451679 10 5 2 3 44091 10 6 1 3 4172 12 1 1 10 62 10 1 7 2 698 12 1 2 9 1976 10 2 6 2 10434 12 2 1 9 1976 10 3 5 2 44091 12 1 3 8 19694 10 4 4 2 69790 12 2 2 8 48846 10 5 3 2 44091 12 3 1 8 19694 10 6 2 2 10434 12 1 4 7 82652 10 7 1 2 698 12 2 3 7 367645 1 o 33 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 263 12 3 2 7 367645 13 3 4 6 20047636 14 1 6 7 2760990 12 4 1 7 82652 13 4 3 6 20047636 14 2 5 7 26837442 12 1 5 6 165262 13 5 2 6 7221592 14 3 4 7 73967488 12 2 4 6 1147628 13 6 1 6 731808 14 4 3 7 73967488 12 3 3 6 2058329 13 1 7 5 533610 14 5 2 7 26837442 12 4 2 6 1147628 13 2 6 5 7221592 14 6 1 7 2760990 12 5 1 6 165262 13 3 5 5 28831218 14 1 7 6 2760990 12 1 6 5 165262 13 4 4 5 44800675 14 2 6 6 36580432 12 2 5 5 1660331 13 5 3 5 28831218 14 3 5 6 144298902 12 3 4 5 4649379 13 6 2 5 7221592 14 4 4 6 223353280 12 4 3 5 4649379 13 7 1 5 533610 14 5 3 6 144298902 12 5 2 5 1660331 13 1 8 4 203280 14 6 2 6 36580432 12 6 1 5 165262 13 2 7 4 3661896 14 7 1 6 2760990 12 1 7 4 82652 13 3 6 4 20047636 14 1 8 5 1533950 12 2 6 4 1147628 13 4 5 4 44800675 14 2 7 5 26837442 12 3 5 4 4649379 13 5 4 4 44800675 14 3 6 5 144298902 12 4 4 4 7259140 13 6 3 4 20047636 14 4 5 5 319684549 12 5 3 4 4649379 13 7 2 4 3661896 14 5 4 5 319684549 12 6 2 4 1147628 13 8 1 4 203280 14 6 3 5 144298902 12 7 1 4 82652 13 1 9 3 38115 14 7 2 5 26837442 12 1 8 3 19694 13 2 8 3 898051 14 8 1 5 1533950 12 2 7 3 367645 13 3 7 3 6542368 14 1 9 4 460245 12 3 6 3 2058329 13 4 6 3 20047636 14 2 8 4 10419653 12 4 5 3 4649379 13 5 5 3 28831218 14 3 7 4 73967488 12 5 4 3 4649379 13 6 4 3 20047636 14 4 6 4 223353280 12 6 3 3 2058329 13 7 3 3 6542368 14 5 5 4 319684549 12 7 2 3 367645 13 8 2 3 898051 14 6 4 4 223353280 12 8 1 3 19694 13 9 1 3 38115 14 7 3 4 73967488 12 1 9 2 1976 13 1 10 2 3080 14 8 2 4 10419653 12 2 8 2 48846 13 2 9 2 94281 14 9 1 4 460245 12 3 7 2 367645 13 3 8 2 898051 14 1 10 3 69765 12 4 6 2 1147628 13 4 7 2 3661896 14 2 9 3 2020530 12 5 5 2 1660331 13 5 6 2 7221592 14 3 8 3 18554641 12 6 4 2 1147628 13 6 5 2 7221592 14 4 7 3 73967488 12 7 3 2 367645 13 7 4 2 3661896 14 5 6 3 144298902 12 8 2 2 48846 13 8 3 2 898051 14 6 5 3 144298902 12 9 1 2 1976 13 9 2 2 94281 14 7 4 3 73967488 12 1 10 1 62 13 10 1 2 3080 14 8 3 3 18554641 12 2 9 1 1976 13 1 11 1 77 14 9 2 3 2020530 12 3 8 1 19694 13 2 10 1 3080 14 10 1 3 69765 12 4 7 1 82652 13 3 9 1 38115 14 1 11 2 4659 12 5 6 1 165262 13 4 8 1 203280 14 2 10 2 172040 12 6 5 1 165262 13 5 7 1 533610 14 3 9 2 2020530 12 7 4 1 82652 13 6 6 1 731808 14 4 8 2 10419653 12 8 3 1 19694 13 7 5 1 533610 14 5 7 2 26837442 12 9 2 1 1976 13 8 4 1 203280 14 6 6 2 36580432 12 10 1 1 62 13 9 3 1 38115 14 7 5 2 26837442 13 10 2 1 3080 14 8 4 2 10419653 12 sum 57167260 13 11 1 1 77 14 9 3 2 2020530 14 10 2 2 172040 13 1 1 11 77 13 sum 438644841 14 11 1 2 4659 13 1 2 10 3080 14 1 12 1 99 13 2 1 10 3080 14 1 1 12 99 14 2 11 1 4659 13 1 3 9 38115 14 1 2 11 4659 14 3 10 1 69765 13 2 2 9 94281 14 2 1 11 4659 14 4 9 1 460245 13 3 1 9 38115 14 1 3 10 69765 14 5 8 1 1533950 13 1 4 8 203280 14 2 2 10 172040 14 6 7 1 2760990 13 2 3 8 898051 14 3 1 10 69765 14 7 6 1 2760990 13 3 2 8 898051 14 1 4 9 460245 14 8 5 1 1533950 13 4 1 8 203280 14 2 3 9 2020530 14 9 4 1 460245 13 1 5 7 533610 14 3 2 9 2020530 14 10 3 1 69765 13 2 4 7 3661896 14 4 1 9 460245 14 11 2 1 4659 13 3 3 7 6542368 14 1 5 8 1533950 14 12 1 1 99 13 4 2 7 3661896 14 2 4 8 10419653 13 5 1 7 533610 14 3 3 8 18554641 14 sum 3369276867 13 1 6 6 731808 14 4 2 8 10419653 13 2 5 6 7221592 14 5 1 8 1533950 264 Ars Math. Contemp. 15 (2018) 147-160 B.3 Genus 2 d v e f H 10 2 5 1 16725 12 1 8 1 7417 5 1 1 1 4 10 3 4 1 47164 12 2 7 1 132202 10 4 3 1 47164 12 3 6 1 721382 5 sum 4 10 5 2 1 16725 12 4 5 1 1610617 10 6 1 1 1649 12 5 4 1 1610617 6 1 1 2 16 12 6 3 1 721382 6 1 2 1 16 10 sum 1355400 12 7 2 1 132202 6 2 1 1 16 11 1 1 7 3633 12 8 1 1 7417 6 sum 48 11 11 1 2 2 1 6 6 50001 50001 12 sum 150429819 7 1 1 3 67 11 1 3 5 201915 13 1 1 9 14091 7 1 2 2 169 11 2 2 5 491729 13 1 2 8 316470 7 2 1 2 169 11 3 1 5 201915 13 2 1 8 316470 7 1 3 1 67 11 1 4 4 315000 13 1 3 7 2241162 7 2 2 1 169 11 2 3 4 1364986 13 2 2 7 5412883 7 3 1 1 67 11 3 2 4 1364986 13 3 1 7 2241162 11 4 1 4 315000 13 1 4 6 6764142 7 sum 708 11 1 5 3 201915 13 2 3 6 28779051 11 2 4 3 1364986 13 3 2 6 28779051 8 1 1 4 237 11 3 3 3 2428862 13 4 1 6 6764142 8 1 2 3 1072 11 4 2 3 1364986 13 1 5 5 9681210 8 2 1 3 1072 11 5 1 3 201915 13 2 4 5 63458654 8 1 3 2 1072 11 1 6 2 50001 13 3 3 5 111801142 8 2 2 2 2664 11 2 5 2 491729 13 4 2 5 63458654 8 3 1 2 1072 11 3 4 2 1364986 13 5 1 5 9681210 8 1 4 1 237 11 4 3 2 1364986 13 1 6 4 6764142 8 2 3 1 1072 11 5 2 2 491729 13 2 5 4 63458654 8 3 2 1 1072 11 6 1 2 50001 13 3 4 4 172252340 8 4 1 1 237 11 1 7 1 3633 13 4 3 4 172252340 11 2 6 1 50001 13 5 2 4 63458654 8 sum 9807 11 3 5 1 201915 13 6 1 4 6764142 11 4 4 1 315000 13 1 7 3 2241162 9 1 1 5 667 11 5 3 1 201915 13 2 6 3 28779051 9 1 2 4 4736 11 6 2 1 50001 13 3 5 3 111801142 9 2 1 4 4736 11 7 1 1 3633 13 4 4 3 172252340 9 1 3 3 8560 13 5 3 3 111801142 9 2 2 3 21113 11 sum 14561360 13 6 2 3 28779051 9 3 1 3 8560 13 7 1 3 2241162 9 1 4 2 4736 12 1 1 8 7417 13 1 8 2 316470 9 2 3 2 21113 12 1 2 7 132202 13 2 7 2 5412883 9 3 2 2 21113 12 2 1 7 132202 13 3 6 2 28779051 9 4 1 2 4736 12 1 3 6 721382 13 4 5 2 63458654 9 1 5 1 667 12 2 2 6 1748723 13 5 4 2 63458654 9 2 4 1 4736 12 3 1 6 721382 13 6 3 2 28779051 9 3 3 1 8560 12 1 4 5 1610617 13 7 2 2 5412883 9 4 2 1 4736 12 2 3 5 6908644 13 8 1 2 316470 9 5 1 1 667 12 3 2 5 6908644 13 1 9 1 14091 12 4 1 5 1610617 13 2 8 1 316470 9 sum 119436 12 1 5 4 1610617 13 3 7 1 2241162 12 2 4 4 10702449 13 4 6 1 6764142 10 1 1 6 1649 12 3 3 4 18938994 13 5 5 1 9681210 10 1 2 5 16725 12 4 2 4 10702449 13 6 4 1 6764142 10 2 1 5 16725 12 5 1 4 1610617 13 7 3 1 2241162 10 1 3 4 47164 12 1 6 3 721382 13 8 2 1 316470 10 2 2 4 115478 12 2 5 3 6908644 13 9 1 1 14091 10 3 1 4 47164 12 3 4 3 18938994 10 1 4 3 47164 12 4 3 3 18938994 13 sum 1506841872 10 2 3 3 206895 12 5 2 3 6908644 10 3 2 3 206895 12 6 1 3 721382 14 1 1 10 25405 10 4 1 3 47164 12 1 7 2 132202 14 1 2 9 700045 10 1 5 2 16725 12 2 6 2 1748723 14 2 1 9 700045 10 2 4 2 115478 12 3 5 2 6908644 14 1 3 8 6235526 10 3 3 2 206895 12 4 4 2 10702449 14 2 2 8 15012496 10 4 2 2 115478 12 5 3 2 6908644 14 3 1 8 6235526 10 5 1 2 16725 12 6 2 2 1748723 14 1 4 7 24417030 10 1 6 1 1649 12 7 1 2 132202 14 2 3 7 103175785 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 B. d 7 7 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 A. Giorgetti and T. R. S. Walsh: Enumeration of hypermaps of a given genus 265 3 2 7 103175785 14 5 3 4 1173398706 14 7 3 2 103175785 4 1 7 24417030 14 6 2 4 306159286 14 8 2 2 15012496 1 5 6 47238510 14 7 1 4 24417030 14 9 1 2 700045 2 4 6 306159286 14 1 8 3 6235526 14 1 10 1 25405 3 3 6 537417269 14 2 7 3 103175785 14 2 9 1 700045 4 2 6 306159286 14 3 6 3 537417269 14 3 8 1 6235526 5 1 6 47238510 14 4 5 3 1173398706 14 4 7 1 24417030 1 6 5 47238510 14 5 4 3 1173398706 14 5 6 1 47238510 2 5 5 435785878 14 6 3 3 537417269 14 6 5 1 47238510 3 4 5 1173398706 14 7 2 3 103175785 14 7 4 1 24417030 4 3 5 1173398706 14 8 1 3 6235526 14 8 3 1 6235526 5 2 5 435785878 14 1 9 2 700045 14 9 2 1 700045 6 1 5 47238510 14 2 8 2 15012496 14 10 1 1 25405 1 7 4 24417030 14 3 7 2 103175785 2 6 4 306159286 14 4 6 2 306159286 14 sum 147326131 3 5 4 1173398706 14 5 5 2 435785878 4 4 4 1799940644 14 6 4 2 306159286 Genus 3 v e f H 13 1 7 1 668591 1 1 1 30 11 sum 10260804 13 2 6 1 8342532 13 3 5 1 31916775 sum 30 12 1 1 6 220244 13 4 4 1 48937240 12 1 2 5 2054974 13 5 3 1 31916775 1 1 2 385 12 2 1 5 2054974 13 6 2 1 8342532 1 2 1 385 12 1 3 4 5567550 13 7 1 1 668591 2 1 1 385 12 2 2 4 13314231 12 3 1 4 5567550 13 sum 2195431068 sum 1155 12 1 4 3 5567550 12 2 3 3 23377106 14 1 1 8 1824323 1 1 3 2900 12 3 2 3 23377106 14 1 2 7 29267487 1 2 2 7070 12 4 1 3 5567550 14 2 1 7 29267487 2 1 2 7070 12 1 5 2 2054974 14 1 3 6 149721473 1 3 1 2900 12 2 4 2 13314231 14 2 2 6 355267058 2 2 1 7070 12 3 3 2 23377106 14 3 1 6 149721473 3 1 1 2900 12 4 2 2 13314231 14 1 4 5 324171185 12 5 1 2 2054974 14 2 3 5 1338142324 sum 29910 12 1 6 1 220244 14 3 2 5 1338142324 12 2 5 1 2054974 14 4 1 5 324171185 1 1 4 15308 12 3 4 1 5567550 14 1 5 4 324171185 1 2 3 65972 12 4 3 1 5567550 14 2 4 4 2041388556 2 1 3 65972 12 5 2 1 2054974 14 3 3 4 3551405485 1 3 2 65972 12 6 1 1 220244 14 4 2 4 2041388556 2 2 2 159608 14 5 1 4 324171185 3 1 2 65972 12 sum 156469887 14 1 6 3 149721473 1 4 1 15308 14 2 5 3 1338142324 2 3 1 65972 13 1 1 7 668591 14 3 4 3 3551405485 3 2 1 65972 13 1 2 6 8342532 14 4 3 3 3551405485 4 1 1 15308 13 2 1 6 8342532 14 5 2 3 1338142324 13 1 3 5 31916775 14 6 1 3 149721473 sum 601364 13 2 2 5 76003367 14 1 7 2 29267487 13 3 1 5 31916775 14 2 6 2 355267058 1 1 5 63355 13 1 4 4 48937240 14 3 5 2 1338142324 1 2 4 418810 13 2 3 4 203571133 14 4 4 2 2041388556 2 1 4 418810 13 3 2 4 203571133 14 5 3 2 1338142324 1 3 3 740100 13 4 1 4 48937240 14 6 2 2 355267058 2 2 3 1779193 13 1 5 3 31916775 14 7 1 2 29267487 3 1 3 740100 13 2 4 3 203571133 14 1 8 1 1824323 1 4 2 418810 13 3 3 3 355620834 14 2 7 1 29267487 2 3 2 1779193 13 4 2 3 203571133 14 3 6 1 149721473 3 2 2 1779193 13 5 1 3 31916775 14 4 5 1 324171185 4 1 2 418810 13 1 6 2 8342532 14 5 4 1 324171185 1 5 1 63355 13 2 5 2 76003367 14 6 3 1 149721473 2 4 1 418810 13 3 4 2 203571133 14 7 2 1 29267487 3 3 1 740100 13 4 3 2 203571133 14 8 1 1 1824323 4 2 1 418810 13 5 2 2 76003367 5 1 1 63355 13 6 1 2 8342532 14 sum 28897471080 266 Ars Math. Contemp. 15 (2018) 147-160 B.5 Genus 4 d v e f H 12 3 1 2 6268712 9 1 1 1 900 12 1 4 1 1510846 14 1 1 6 38547144 12 2 3 1 6268712 14 1 2 5 338317960 9 sum 900 12 3 2 1 6268712 14 2 1 5 338317960 12 4 1 1 1510846 14 1 3 4 890383128 10 1 1 2 19344 14 2 2 4 2093639428 10 1 2 1 19344 12 sum 57017238 14 3 1 4 890383128 10 2 1 1 19344 14 1 4 3 890383128 13 1 1 5 8417332 14 2 3 3 3622371084 10 sum 58032 13 1 2 4 52864504 14 3 2 3 3622371084 13 2 1 4 52864504 14 4 1 3 890383128 11 1 1 3 207876 13 1 3 3 91902888 14 1 5 2 338317960 11 1 2 2 496224 13 2 2 3 216973192 14 2 4 2 2093639428 11 2 1 2 496224 13 3 1 3 91902888 14 3 3 2 3622371084 11 1 3 1 207876 13 1 4 2 52864504 14 4 2 2 2093639428 11 2 2 1 496224 13 2 3 2 216973192 14 5 1 2 338317960 11 3 1 1 207876 13 3 2 2 216973192 14 1 6 1 38547144 13 4 1 2 52864504 14 2 5 1 338317960 11 sum 2112300 13 1 5 1 8417332 14 3 4 1 890383128 13 2 4 1 52864504 14 4 3 1 890383128 12 1 1 4 1510846 13 3 3 1 91902888 14 5 2 1 338317960 12 1 2 3 6268712 13 4 2 1 52864504 14 6 1 1 38547144 12 2 1 3 6268712 13 5 1 1 8417332 12 1 3 2 6268712 14 sum 24635879496 12 2 2 2 14872428 13 sum 1269067260 B.6 Genus 5 d v e f H 13 1 1 3 22482432 14 2 1 3 853365360 11 1 1 1 54990 13 1 2 2 52815168 14 1 3 2 853365360 13 2 1 2 52815168 14 2 2 2 1995345826 11 sum 54990 13 1 3 1 22482432 14 3 1 2 853365360 13 2 2 1 52815168 14 1 4 1 211558928 12 1 1 2 1588218 13 3 1 1 22482432 14 2 3 1 853365360 12 1 2 1 1588218 14 3 2 1 853365360 12 2 1 1 1588218 13 sum 225892800 14 4 1 1 211558928 12 sum 4764654 14 1 1 4 211558928 14 sum 7750214770 14 1 2 3 853365360 B.7 Genus 6 d v e f H 13 1 11 5263764 14 1 1 2 192834612 14 sum 578503836 14 1 2 1 192834612 13 sum 5263764 14 2 1 1 192834612