Also available at http://amc.imfm.si
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ARS MATHEMATICA CONTEMPORANEA 4 (2011) 351–361
Enumeration of genus–four maps
by number of edges
Alexander Mednykh ∗
Sobolev Institute of Mathematics, Novosibirsk State University
630090 Novosibirsk, Russia
Alain Giorgetti
LIFC, University of Franche-Comté
16 route de Gray, 25030 Besançon CEDEX, France
Received 10 December 2009, accepted 4 August 2011, published online 5 October 2011
Abstract
An explicit form of the ordinary generating function for the number of rooted maps
on a closed orientable surface of genus four with a given number of edges is given. An
analytical formula for the number of unrooted maps of genus four with a given number of
edges is obtained through the number of rooted ones. Both results are new.
Keywords: Enumeration, surface, orbifold, rooted map, unrooted map, Fuchsian group.
Math. Subj. Class.: 05C30, 57R18, 20H10
1 Introduction
By a map we mean a 2-cell decomposition of a compact orientable surface. A map is rooted
if one of its darts (edge–vertex incidence pairs) is distinguished as a root and unrooted oth-
erwise. Enumeration of maps up to orientation–preserving homeomorphism has attracted
a lot of attention in the last decades. Enumeration of rooted maps on the plane by number
of edges was done in the pioneering paper by W. T. Tutte [18]. Later, an analogous result
for the torus was given by D. Arquès [2]. A structure for the generating function of the
number of rooted maps on the surface of given genus g was suggested by E. A. Bender and
E. R. Canfield [5]. By making use of their approach they obtained numerical tables of num-
bers of rooted maps for genera 2 and 3. The enumeration of rooted maps of given genus by
∗Supported by the RFBR (grant 06-01-00153), APVV SK-RU-0007-07, FCPK (grant 02.740.11.0457).
E-mail addresses: mednykh@math.nsc.ru (Alexander Mednykh), alain.giorgetti@univ-fcomte.fr (Alain
Giorgetti)
Copyright c© 2011 DMFA Slovenije
352 Ars Math. Contemp. 4 (2011) 351–361
number of vertices and faces was done in [3]. Also, a general algorithm to compute the gen-
erating function for such numbers was suggested. The results of this algorithm confirm the
available data [22] obtained earlier by T. Walsh and A. Lehman for small numbers of edges.
Enumeration of unrooted maps is a more complicated problem. For unrooted maps on the
oriented sphere a closed analytical counting formula (of the type studied in the present pa-
per) was given by V. A. Liskovets [12]. Counting algorithms suitable for maps either on
the oriented sphere or on the general one were independently provided by N. C. Wormald
[23, 24]. A general formula for counting unrooted maps through the number of rooted ones
was obtained in [14]. In particular, the problem was completely solved for the torus and
the orientable surfaces of genera 2 and 3. Different approaches to counting unrooted maps
with prescribed properties were given in the papers [13, 8, 20].
The present paper provides two new results for maps on a closed orientable surface
of genus 4. The first result is an explicit form of the ordinary generating function for the
number of rooted maps of genus 4 with a given number of edges. The second result is an
explicit analytical formula relating the number of unrooted maps of genus 4 with n edges
with numbers of rooted maps of genus g ≤ 4 and with e ≤ n edges.
2 Orientable combinatorial maps
By a (combinatorial) map we mean a triple (D;R,L) composed of a finite set D and two
permutations R and L, with L satisfying L2 = 1, generating a transitive subgroup of the
symmetric group SD. The elements of D are called darts and the orbits of R, L and RL
are respectively called vertices, edges and faces. Edges of size one are called semiedges.
The genus g of a map M = (D;R,L) is given by 2 − 2g = V + E + F − |D|, where V
is the number of vertices, E is the number of edges and F is the number of faces. If M has
no semiedges (i.e. if L is fixed–point free), then |D| = 2E and 2− 2g = V − E + F .
Combinatorial maps describe topological maps on orientable surfaces with a chosen
global orientation; the permutation R represents the cyclic order of the edge-ends incident
with each vertex encountered by a rotation around that vertex in the direction correspond-
ing to that orientation. Hence they are determined up to orientation–preserving homeo-
morphisms of the surface leaving the set of vertices, of edges and of faces invariant. This
gives rise to the following definition: two maps (D1;R1, L1) and (D2;R2, L2) are isomor-
phic, written (D1;R1, L1) ∼= (D2;R2, L2), if there is a bijection ψ : D1 → D2 such that
ψR1 = R2ψ and ψL1 = L2ψ. The isomorphism class of the map (D;R,L) is the set of
maps (D;Rψ, Lψ) such that ψ ∈ SD is a permutation of D. A rooted map is a 4-tuple
(D, r;R,L), where r ∈ D and (D;R,L) is a map. The dart r is called the root. Two
rooted maps (D, r;R,L) and (D′, r′;R′, L′) are isomorphic is there is an isomorphism
(D,R,L) onto (D′, R′, L′) taking root r to root r′. To each map M = (D;R,L) there
is an associated closed orientable surface (that is, a compact orientable surface without
boundary) which can be constructed by associating a 2-cell to each orbit of the permutation
RL. Hence M can be regarded as a topological map. In turn, any topological map on a
closed orientable surface can be realized as a combinatorial map and two topological maps
are related by an orientation-preserving homeomorphism if and only if the corresponding
combinatorial maps are related by an isomorphism [10].
By enumerating unrooted maps we mean enumerating isomorphism classes of com-
binatorial maps, which is equivalent to enumerating topological maps up to orientation-
preserving homeomorphism. Topological maps correspond to cellular embeddings of
A. Mednykh and A. Giorgetti: Enumeration of genus–four maps by number of edges 353
graphs. Since graphs were generally assumed to be without semiedges, we will interpret
“maps on a closed orientable surface of genus g”, or just “genus g orientable maps” as
maps without semiedges.
The theory of maps presented in [10] gives a close relationship between maps and
subgroups of a certain universal group. Denote by ∆ = ∆(∞,∞, 2) the group 〈α, β|β2 =
1〉 ∼= Z ∗ Z2. Given a map (D;R,L), the assignment α 7→ R and β 7→ L extends to an
epimorphism Φ : ∆ → 〈R,L〉. It follows that ∆ acts on D by z · x = Φ(z)x for z ∈ ∆
and x ∈ D. The stabilizer K ≤ ∆ of a dart x ∈ D has index [∆ : K] = |D|. Conversely,
each subgroup K ≤ ∆ of finite index determines a rooted map M = (D, r;R,L), where
D is the set of left cosets xK, x ∈ ∆, r = K is the trivial coset and the action of R and L
is defined by left multiplication: R(xK) = αxK, L(xK) = βxK. Moreover, M has no
semiedges if and only if K is torsion–free.
2.1 Maps on orbifolds
In this paper we consider maps on orbifolds. This is a new and fruitful idea already
used in previous papers [14, 15]. By an oriented orbifold O we mean an oriented sur-
face S with a discrete subset of points B = {p1, p2, . . .} such that to each point pi an
integer mi ≥ 2 is assigned. The elements of B will be called branch points and the
respective numbers m1,m2, . . . ,mi, . . . will be called branch indices. If S is a com-
pact connected orientable surface of genus g, then B is of finite cardinality |B| = r. In
this case, the orbifold O is uniquely determined (up to orientation–preserving homeomor-
phism) by its signature [g;m1,m2, . . . ,mr], 1 < m1 ≤ m2 ≤ . . . ≤ mr. Hence we write
O = O[g;m1,m2, . . . ,mr]. The fundamental group π1(O) of O is an F -group (see [10])
defined by
π1(O) = F [g;m1,m2, . . . ,mr]
= 〈a1, b1, a2, b2, . . . , ag, bg, e1, . . . , er|
g∏
i=1
[ai, bi]
r∏
j=1
ej = 1, e
m1
1 = · · · = emrr = 1〉. (2.1)
A map on an orbifold O is a map on the underlying surface Sg of genus g satisfying the
following three properties:
(P1) if x ∈ B, then x is either an internal point of a face, or a vertex, or an end-point of a
semiedge (free end) which is not a vertex,
(P2) each face contains at most one branch point,
(P3) each free end of a semiedge is a branch point and the branch index of this point is 2.
Maps on orbifolds arise naturally when we take a quotient of an ordinary map on a
closed surface by a finite group G of automorphisms. Then the numbers m1, . . . ,mr are
the orders of the stabilizers of the faces, vertices and edges under the action ofG. Note that
these stabilizers are always cyclic. Further information on maps on orbifolds can be found
in [14] and [15].
An epimorphism π1(O) → Z` onto a cyclic group of order ` is called order pre-
serving if it preserves the orders of the generators ej , j = 1, . . . , r. Equivalently, an
order–preserving epimorphism π1(O) → Z` has a torsion–free kernel. We denote by
Epi0(π1(O), Z`) the number of order–preserving epimorphisms π1(O)→ Z`.
354 Ars Math. Contemp. 4 (2011) 351–361
For a technical reason it is convenient to modify the signature in the following way. Let
[g;m1,m2, . . . ,mr] = [g; 2, . . . , 2︸ ︷︷ ︸
b2 times
, 3, . . . , 3︸ ︷︷ ︸
b3 times
, . . . , `, . . . , `︸ ︷︷ ︸
b` times
].
Then we write [g; 2b2 , 3b3 , . . . , `b` ] rather than [g;m1,m2, . . . ,mr] listing only those jbj
with bj > 0.
Denote by Orb(Sg/Z`) the set of `-tuples [g; 2b2 , 3b3 , . . . , `b` ] which are the signatures
of cyclic orbifolds of type Sg/Z` for some Sg and Z`. By definition, the fundamental
group π1(O) is uniquely determined by the signature of the orbifold O. Hence, for any
O ∈ Orb(Sg/Z`), O = [g; 2b2 , 3b3 , . . . , `b` ], the group π1(O) is well defined.
3 Rooted map enumeration
Let Qg(z) =
∑
n≥0Ng(n)z
n be the ordinary generating function counting the number
Ng(n) of rooted maps on the orientable surface of genus g by number of edges (the ex-
ponent of z). This generating function satisfies an equation system presented by E. A.
Bender and E. R. Canfield in [4]. Given a genus g, a closed expression for Qg(z) can in
principle be computed from this equation system by induction. However the computational
complexity is so high that up to 1998 exact solutions where only known for the first four
genera, from 0 to 3. A common pattern for all the Qg(z), where g ranges over the positive
integers, was proposed in [5]. EachQg(z) is a rational function of a quadratic parameter of
z, but this pattern leaves a polynomial of this parameter unknown. An upper bound for the
polynomial degree is conjectured but not proved. The first proof of a more precise pattern,
with a maximal degree for each unknown polynomial, is due to D. Arquès and the second
author [3, 9], for the more general case of counting by number of vertices and faces. This
section presents new results derived from this former work by focusing on counting by
number of edges.
3.1 Generating functions counting rooted maps
The following result proves the conjecture in [5] and is an easy consequence of Theorem 1
of [3].
Theorem 3.1. For any positive integer g, the ordinary generating function Qg(z) counting
rooted maps on a closed orientable surface of genus g by number of edges (exponent of z)
can be written
Qg(z) = z
2g(1− 3m)−2(1− 2m)4−5g(1− 6m)3−5gPg(m),
where m =
1−
√
1− 12z
6
and Pg(m) is a polynomial of m of degree less than or equal
to 6g − 6.
The explicit formulae for polynomials Pg(m), g = 0, 1, 2, 3 can be derived from the
papers [19], [2], [5] and [9], respectively. The first step to counting unrooted genus–four
maps by number of edges is given by the following proposition.
Proposition 3.2. The polynomial Pg in Theorem 3.1 for g = 4 is given by the formula
A. Mednykh and A. Giorgetti: Enumeration of genus–four maps by number of edges 355
P4(m) = 9(1− 2m)6
(41956066368m12 − 107657028288m11 + 128766120048m10
−95026128096m9 + 48202134300m8 − 17709582732m7
+4855070265m6 − 1025233956m5 + 178608786m4
−28633200m3 + 4245462m2 − 465894m+ 25025).
(3.1)
Proof. Theorem 5.1, Relation (6.1) and Propositions 6.1 and 6.2 from [9] make it possible
to compute Pg from polynomials of lower degree. This result has been computed by a
software developed by the second author in his Ph.D. thesis. The software does not directly
compute the polynomials but rather the generating functions enumerating rooted maps by
genus, along the principles presented in [5] and detailed in [9]. It has not been designed for
efficiency but it successfully computes the generating functions for rooted maps up to genus
4. The genus–four formula was computed in less than one minute on an Intel Pentium at 1.4
GHz with 1.25 Gb of memory. In accordance with Theorem 3.1, the polynomial computed
for P4 is indeed of degree 6g − 6 = 18.
3.2 Counting rooted genus–four maps by number of edges
Theorem 3.3. The ordinary generating function Q4(z) counting rooted maps on a closed
orientable surface of genus four by number of edges (the exponent of z) is
Q4(z) = 9z
8(1− 3m)−2(1− 2m)−10(1− 6m)−17
(41956066368m12 − 107657028288m11 + 128766120048m10
−95026128096m9 + 48202134300m8 − 17709582732m7
+4855070265m6 − 1025233956m5 + 178608786m4
−28633200m3 + 4245462m2 − 465894m+ 25025),
(3.2)
where m =
1−
√
1− 12z
6
.
Proof. Formula (3.2) follows from Theorem 3.1 and Proposition 3.2.
Table 1 presents the coefficients of Qg(z) =
∑
n≥0Ng(n)z
n. The first seven coef-
ficients correspond to the ones found by Timothy Walsh in his Ph.D. thesis [21]. The
remaining coefficients are new.
4 Unrooted map enumeration
4.1 Counting unrooted maps through rooted maps on orbifolds
The following theorem is the main result of [14].
Theorem 4.1. The number Ug(e) of unrooted maps with e edges on a closed orientable
surface of genus g is given by the formula
Ug(e) =
1
2e
∑
`m=2e
∑
O∈Orb(Sg/Z`)
Epi0(π1(O), Z`)νO(m),
where νO(m) is the number of rooted maps with m darts on the orbifold O.
356 Ars Math. Contemp. 4 (2011) 351–361
n The number N4(n) of rooted maps of genus 4 with n edges
8 225225
9 24635754
10 1495900107
11 66519597474
12 2416610807964
13 75981252764664
14 2141204115631518
15 55352670009315660
16 1334226671709010578
17 30347730709395639732
18 657304672067357799042
19 13652607304062788395788
20 273469313030628783700080
21 5306599156694095573465824
22 100128328831437989131706976
23 1842794650155970906232185656
24 33167202398202989127880734894
25 585079650671639944950451625580
26 10134917623511547808118654370114
27 172678013694177771071548169002188
28 2897912714075648947715005321906392
29 47963145773909943419634526762950192
30 783757995914247522485178250636927380
31 12657015244648210693716700196736399336
32 202177082281879102698899470748726765316
33 3196834110175421253323791465873251739560
34 50072299181065185108291501010224952255668
35 777384663760023780739632793721755383049272
36 11969638731261482998116895312223651253180480
37 182875502596501323216343759769794526714561664
38 2773716775724835345230901154059649970954877396
39 41781661724286164921640221635213792280118832368
40 625310196714095279935937237998816771771464314790
41 9301365625304817339752604766781541863133507845340
42 137556789724353166312824029682866215741796911453698
43 2023172807939725017933640132814869413798020476575564
44 29601998835280343256197223863418277211551813053748872
45 430981509422356688373368386557125320381885703792230800
Table 1: Enumeration of rooted maps of genus 4 by number of edges.
The number of rooted maps on the orbifold O = O[g; 2b2 , . . . , `b` ] can be expressed
through the number Ng(n) of rooted maps on genus g surface by the following proposition
given in [14].
Proposition 4.2. Let O = O[g; 2b2 , . . . , `b` ] be an orbifold, bi ≥ 0 for i = 2, . . . , `. Let
A. Mednykh and A. Giorgetti: Enumeration of genus–four maps by number of edges 357
Ng(n) be the number of rooted maps of genus g with n edges. Then the number of rooted
maps νO(m) with m darts on the orbifold O is
νO(m) =
b2∑
s=0
(
m
s
)( m−s
2 +2−2g
b2−s, b3, . . . , b`
)
Ng((m−s)/2),
where Ng(n) = 0 if n is not an integer.
Denote by µ(n), φ(n) and Φ(x, n) the Möbius, Euler and von Sterneck functions [1,
16]. The relationship between them is given by the formula
Φ(x, n) =
φ(n)
φ( n(x,n) )
µ
(
n
(x, n)
)
,
where (x, n) is the greatest common divisor of x and n. It was shown by O. Hölder that
Φ(x, n) coincides with the Ramanujan sum
∑
1≤k≤n
(k, n)=1
exp( 2 ikxn ). For the proof, see [1, p.164]
and [16].
Recall that the Jordan multiplicative function φk(n) of order k can be defined (for more
information see [7, p.199], [11, 17]) as
φk(n) =
∑
d|n
µ
(n
d
)
dk.
From [14] we have the following proposition.
Proposition 4.3. Let O = O[g;m1, . . . ,mr] be an orbifold of signature (g;m1, . . . ,mr).
Denote by m = lcm(m1, . . . ,mr) the least common multiple of m1, . . . ,mr and let `
be a multiple of m. Then the number of order-preserving epimorphisms of the orbifold
fundamental group π1(O) onto a cyclic group Z` is given by the formula
Epi0(π1(O), Z`) = m
2gφ2g(`/m)E(m1,m2, . . . ,mr),
where
E(m1,m2, . . . ,mr) =
1
m
m∑
k=1
Φ(k, m1) · Φ(k, m2) · . . . · Φ(k, mr),
φ2g(`) is the Jordan multiplicative function of order 2g, and Φ(k, m) is the von Sterneck
function.
358 Ars Math. Contemp. 4 (2011) 351–361
4.2 Counting unrooted genus–four maps by number of edges
Theorem 4.4. The number U4(e) of unrooted maps on a closed orientable surface of genus
four counted by the number of edges e is given by the formula
1
2e
(
N4(e) + 16ν[2;22](e) + 4ν[1;26](e) + ν[0,210](e)
+80N2(e/3) + 18ν[1;33](2e/3) + 22ν[0;36](2e/3)
+32ν[1;42](e/2) + 8ν[0;2,44](e/2) + 2ν[0;24,42](e/2)
+52ν[0;54](2e/5) + 32ν[1;22](e/3) + 2ν[0;2,63](e/3)
+2ν[0;22,33](e/3) + 6ν[0;32,62](e/3) + 2ν[0;23,3,6](e/3)
+4ν[0;22,82](e/4) + 18ν[0;93](2e/9) + 12ν[0;5,102](e/5)
+4ν[0;22,52](e/5) + 4ν[0;3,122](e/6) + 4ν[0;4,6,12](e/6)
+8ν[0;3,5,15](2e/15) + 8ν[0;2,162](e/8) + 6ν[0;2,9,18](e/9)
)
,
where νO(m) is defined in Proposition 4.2 and Ng(e) is the number of rooted maps of
genus g with e edges.
Proof. O.V. Bogopol’skii [6] described all possible signatures of orbifolds of the type
S4/G, where S4 is a surface of genus four and G is a finite group of homeomorphisms act-
ing on S4. In particular, G = Z` is cyclic of order ` only for ` = 1, 2, 3, 4, 5, 6, 8, 9, 10,
12, 15, 16 and 18.
From this observation and Proposition 4.3 we get the following lemma.
Lemma 4.5. Let O ∈ Orb(S4/Z`). Then one of the following cases occurs:
(1) ` = 1 : O = O[4; ∅] with Epi0(π1(O), Z`) = 1;
(2) ` = 2 : O = O[2; 22], O[1; 26], O[0; 210] with Epi0(π1(O), Z`) = 16, 4, 1;
(3) ` = 3 : O = O[2; ∅], O[1; 33], O[0; 36] with Epi0(π1(O), Z`) = 80, 18, 22;
(4) ` = 4 : O = O[1; 42], O[0; 2, 44], O[0; 24, 42] with Epi0(π1(O), Z`) = 32, 8, 2;
(5) ` = 5 : O = O[0; 54] with Epi0(π1(O), Z`) = 52;
(6) ` = 6 : O = O[1; 22], O[0; 2, 63], O[0; 22, 33], O[0; 32, 62], O[0; 23, 3, 6]
with Epi0(π1(O), Z`) = 32, 2, 2, 6, 2;
(7) ` = 8, 9 : O = O[0; 22, 82], O[0; 93] with Epi0(π1(O), Z`) = 4, 18;
(8) ` = 10 : O = O[0; 5, 102], O[0; 22, 52] with Epi0(π1(O), Z`) = 12, 4;
(9) ` = 12 : O = O[0; 3, 122], O[0; 4, 6, 12] with Epi0(π1(O), Z`) = 4, 4;
(10) ` = 15, 16, 18 : O = O[0; 3, 5, 15], O[0; 2, 162], O[0; 2, 9, 18]
with Epi0(π1(O), Z`) = 8, 8, 6.
Now, the theorem follows from Lemma 4.5 and Theorem 4.1.
We present the numbers thus obtained in Table 2.
A. Mednykh and A. Giorgetti: Enumeration of genus–four maps by number of edges 359
n The number U4(n) of unrooted maps of genus 4 with n edges
8 14118
9 1369446
10 74803564
11 3023693380
12 100692692173
13 2922359760376
14 76471600288836
15 1845089145736960
16 41694584320696782
17 892580319444417876
18 18258463136626650660
19 359279139700128276168
20 6836732826365623258492
21 126347598971804884131800
22 2275643837092089686415858
23 40060753264325317709454720
24 690983383296198882647616692
25 11701593013434174490416914028
26 194902261990612930685627941344
27 3197740994336653065511697474864
28 51748441322779568341478022803550
29 826950789205344386488852660387184
30 13062633265237461036677963280146184
31 204145407171745343738289312062076704
32 3159016910654361022421358641441865404
33 48436880457203352503593806713722630064
34 736357340898017428826654622692598290184
35 11105495196571768299465860194739273233104
36 166244982378631708320492461115910055656280
37 2471290575628396259735279442756616067154240
38 36496273364800465069054923670966215780880134
39 535662329798540575919393860220780811735355616
40 7816377458926190999203023224069385048268971894
41 113431288113473382192120378978151871847586693068
42 1637580830051823408486063085325625200796356325376
43 23525265208601453696903044755726884736237505268568
44 336386350400912991547696743715303742977503309390766
45 4788683438026185426370763920603749547029983530699104
Table 2: Enumeration of unrooted maps of genus 4 by number of edges.
5 Acknowledgments
The idea of writing this paper was born as a result of a discussion between Valery Liskovets,
Roman Nedela, Timothy Walsh and the first author during the conference GEMS’09 held
in Tále, Slovakia, 28 June – 03 July, 2009. We thank Prof. T. Walsh for his stimulating
360 Ars Math. Contemp. 4 (2011) 351–361
discussions by e-mail and for his corrections of an early version of the present text. The
authors are also grateful to the anonymous referees for helpful comments and suggestions.
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