Acta Mathematica Sinica, English Series Jun., 2008, Vol. 24, No. 6, pp. 937–946 Published online: June 4, 2008 DOI: 10.1007/s10114-007-6213-x Http://www.ActaMath.com
Acta Mathematica Sinica, English Series The Editorial Office of AMS & Springer-Verlag 2008
A Census of Petal Bundles by Genus Yan XU School of Science, Tianjin Polytechnic University, Tianjin 300160, P. R. China E-mail: xu yanr@ 163.com
Yan Pei LIU Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China E-mail:
[email protected] Abstract This paper provides the uniform enumerative functional equation of orientable (nonorientable) rooted petal bundles with more parameters, and deduces two recursion formulas for calculation. Accordingly, an explicit expression of rooted petal bundles with up to two parameters on nonorientable surface of genus 4 is also obtained. Keywords
rooted map, enumerating function, enumeration, petal bundles
MR(2000) Subject Classification 05C10, 05C30
1
Introduction
A surface is a compact 2-manifold without boundary. Surfaces with p handles, i.e., Op (q crosscaps, i.e., Nq ) are called orientable (nonorientable) of genus p (q). A map M = (χα,β , J ) is defined to be a basic permutation J , i.e., for any x ∈ χα,β , no integer k exists such that J k x = αx, acting on χα,β , the disjoint union of quadricells Kx of x ∈ X(the base set), where K = {1, α, β, αβ} is the Klein group, satisfying the following conditions: i) αJ = J −1 α; ii) The group ΨJ generated by J = {α, β, J } is transitive on χ. A map is rooted if an element in χα,β is marked and called the root r. The vertex, edge and face incident with r are, respectively, the root-vertex vr (M ), the root-edge er (M ), and the root-face fr (M ). Two rooted maps M = (χα,β , J ) with root rM and N = (Yα,β , P) with root rN are said to be isomorphic if there is an isomorphism between them such that their roots rM and rN are in correspondence. An edge belonging to only one face is called double (or singular by some author), all others belonging to exactly two faces are called single. Concerning other definitions or denotations of a map on a surface, the reader can refer to the literature, [1] or [2]. A petal bundle (or one-vertexed map) is a map with only one vertex. The concept of rooted map was first introduced by Tutte [3–4] in the 1960’s to attack the Four Color Problem. His series of census papers laid a solid foundation for current enumerative theory. Since then, much work has been done by Tutte himself, Arqu`es, Brown [5], Mullin, Harary, Bender et al. [6], Liu [1–2, 7] et al. in a series of papers. Among them, Arqu`es, Brown, Received April 27, 2006, Accepted July 25, 2007 Supported by NNSFC under Grant No. 60373030 and Grant No. 10571031
Xu Y. and Liu Y. P.
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Walsh, Lehman, Bender et al. and Gao (asymptotic method) et al. did some influential work on the (exact) enumeration of nonplanar maps. Recently, Cai and Liu [8], Ren and Liu [9] (4-regular maps etc.), Li and Liu [10], Hao et al. also obtained some good results for the exact enumeration of maps. It is well known that petal bundles play an important role in the study of embeddings of graphs on surfaces. In this article, on the basis of the theory from the monograph [7], we discuss the orientable (nonorientable) rooted petal bundles in detail. Firstly, in Section 2 and Section 3, on surfaces with arbitrary genus, the uniform enumerative functional equation of orientable (nonorientable) rooted petal bundles with more parameters are obtained. Meanwhile, the recursion formulas about the number of petal bundles are also obtained. Finally, on the basis of the above results, the explicit expression of rooted petal bundles with two parameters on N4 is derived, and as a corollary the number of rooted petal bundles with the size is also obtained. Let Op and Nq be the set of petal bundles on Op and Nq , respectively. Suppose their enumerating functions are, respectively: xm(M ) y n(M ) = Gpm,n xm y n , gp (x, y) = m,n≥0
M ∈Op
fq (x, y) =
x
m(M ) n(M )
y
=
q Fm,n xm y n ,
m,n≥1
M ∈Nq
where p ≥ 0, q ≥ 1, m(M ), n(M ) are the valency of the root-face and the size, respectively. Also, let Dip (p ≥ 0) and Liq (q ≥ 1) be the set of all petal bundles with i (i ≥ 0) distinguished non-rooted faces on Op and Nq , respectively. Their enumerating functions are, respectively: Denote z = (z1 , z2 , . . . , zi ), k = (k1 , k2 , . . . , ki ), then xm(M ) y n(M ) z k(M ) = Dip (x, y, z) = M ∈Dip
Lqi (x, y, z)
=
m,n,k≥0
x
m(M ) n(M ) k(M )
y
z
=
M ∈Liq
m,n,k≥1
dpm,n,k xm y n z k , q lm,n,k xm y n z k ,
p where dpm,n,k and lm,n,k count, respectively, petal bundles in Dip and Liq with the valency of root-face m, the size n and the valency of j-th distinguished non-rooted face kj (kj ≥ 1, j = 1, . . . , i). Apparently, when i = 0, enumerating functions Dip and Lqi are equal to gp (x, y) and fq (x, y), respectively, (p ≥ 0, q ≥ 1). What’s more, Dip = 0, p < 0; Lqi = 0, q ≤ 0. Homoplastically, we have the following definitions: z˜ = (z1 , . . . , zi , zi+1 ), k˜ = (k1 , . . . , ki ,
ki+1 ), z˙ = (z1 , . . . , zj−1 , zj+1 , . . . , zi ) and k˙ = (k1 , . . . , kj−1 , kj+1 , . . . , ki ), ˜ ˜ p (x, y, z˜) = xm(M ) y n(M ) z˜k(M ) = dpm,n,k˜ xm y n z˜k , Di+1 p M ∈Di+1
p (x, y, z) ˙ Di−1
=
p M ∈Di−1
˜ m,n,k≥0
x
˙ ) m(M ) n(M ) k(M
y
z˙
=
˙ m,n,k≥0
˙
dpm,n,k˙ xm y n z˙ k .
˙ their representations are known to be similar to the above. As for Lqi+1 (x, y, z˜) and Lqi−1 (x, y, z),
A Census of Petal Bundles by Genus
Theorem 1.1 [11] sion:
939
The enumerating function g0 (x, y) = D00 has the following explicit expresg0 (x, y) = 1 +
m(2n − m − 1)! xm y n . (n − m)!n!
(1.1)
m,n≥1
In this article, some main results are obtained as follows : ⎧ ⎨ 1, q is odd number ;
Let γq =
⎩ 0, q is even number.
Theorem A The functions Dip (x) = Dip (x, y, z)(p ≥ 0), except for i = p = 0, satisfy the following equation : (1 − x + x2 y)Dip (x) ∗
= xyDip + (1 − x)
i j=1
∗
p xyzj δzj ,x (uDi−1 (u)) + (1 − x)x3 y
p−1 ∂Di+1 (x, x), ∂zi+1
p p zj Di−1 (zj ,y,z)−xD ˙ ˙ i−1 (x,y,z) as (1.6.7) in [2]. zj −x q q function Li (x) = Li (x, y, z)(q ≥ 1) satisfies the
(1.2)
p where Dip = Dip (1), δzj ,x (uDi−1 (u)) =
Theorem B equation :
The enumerating
(1 − x + x2 y)Lqi (x) = (1 − x)
i j=1
xyzj δzj ,x (uLqi−1 (u)) q−1
+ (1 − x)x2 y[(xLq−1 (x)) + γq (xDi 2 (x)) ] + xyLqi i q 2 −1 ∂Lq−2 ∂Di+1 i+1 3 + (1 − x)x y 2 (x, x) + γq+1 (x, x) , ∂zi+1 ∂zi+1 ∗
where Lqi = Lqi (1), δzj ,x (uLqi−1 (u)) = 2
following
∗
(1.3)
q zj Lqi−1 (zj ,y,z)−xL ˙ ˙ i−1 (x,y,z) . zj −x
Orientable Rooted Petal Bundles
Since the exact expression of planar petal bundles is known, here, we discuss Dip (i, p ≥ 0 and they are not zero at the same time) mostly. For Dip (except for i = p = 0), it can be classified into three categories: DipI , DipII and DipIII :
• M ∈ DipI , er (M ) is a single edge and the j-th (j = 1, 2, . . . , i) distinguished non-rooted face is adjacent to the root-edge;
• M ∈ DipII , er (M ) is a single edge and any distinguished non-rooted face is not adjacent to the root-edge; • M ∈ DipIII , er (M ) is a double edge (Remark: when p = 0, this category doesn’t exist). Lemma 2.1
p
p
p . Let Di I = {M − er (M ) | M ∈ DipI }. Then Di I = Di−1
Proof For any M ∈ DipI , by moving the root-edge er (M ), the j-th distinguished non-rooted p p p . Similarly, for any M ∈ Di−1 , face and the root-face of M are merged into one, so Di I ⊆ Di−1 pI M ∈ Di can be obtained by adding a new edge R from the tail of er (M ) to any vertex on the root-face boundary of M as the root-edge of M , which can separate the root-face of M
Xu Y. and Liu Y. P.
940
into two faces—the root-face of M and a new (denoted by j) distinguished non-rooted face. p p . This lemma holds. Therefore, Di I ⊇ Di−1 Lemma 2.2
p
p
Let Di II = {M − er (M ) | M ∈ DipII }. Then Di II = Dip .
Proof The proof is similar to that of Lemma 2.1. Lemma 2.3
p
p
p−1 . Let Di III = {M − er (M ) | M ∈ DipIII }. Then Di III = Di+1
Proof For any M ∈ DipIII , according to the definition of DipIII , the root-face of M is divided into two faces by deleting the double root-edge er (M ). One is the root-face of M = M −er (M ), here er (M ) = J r; the other face can be viewed as the (i+1)-th distinguished non-rooted face of M . Since v(M ) − ε(M ) + φ(M ) = 2 − 2p, it is easy to see that v(M ) − ε(M ) + φ(M ) = 2 − 2(p − 1), p p−1 p−1 . Then Di III ⊆ Di+1 . M ∈ Di+1 p−1 , since v(M ) − ε(M ) + φ(M ) = 2 − 2(p − 1), we On the other hand, for any map M ∈ Di+1 pIII can get M ∈ Di by adding a new double edge R from the tail of the root-edge of M to each of the vertices on the boundary of the (i + 1)-th distinguished non-rooted face as the root-edge of M so that the (i + 1)-th distinguished non-rooted face and the root-face can be turned into a new face whose valency is m(M ) + ki+1 (M ) + 2. Since M = M − R , M ∈ DipIII it is evident p p p−1 . that M ∈ Di III . Therefore, Di III = Di+1 Let Dips (s = 1, 2, 3) be, respectively, the enumerating functions of DipI , DipII and DipIII . Then as a direct consequence of Lemmas 2.1, 2.2 and 2.3, the following formulas hold: Dip1 (x, y, z) =
i
xyzj
p M ∈Di−1
j=1
˙
(zjm + xzjm−1 + · · · + xm )y n z˙ k
i
p p zj Di−1 (zj , y, z) ˙ − xDi−1 (x, y, z) ˙ ; = xyzj z − x j j=1
Dip2 (x, y, z) = y =
(x + x2 + · · · + xm+1 )y n z k
M ∈Dip
∗ xy [Dp − xDip ]; 1−x i
∗
where Dip = Dip (1, y, z).
Dip3 (x, y, z) = y
ki+1 xm+ki+1 +2 y n z k
p−1 M ∈Di+1
= x3 y
p−1 ∂Di+1 (x, y, z, x). ∂zi+1
Since Dip = Dip1 + Dip2 + Dip3 , then the theorem A is true. The uniqueness can be directly derived from the recursion obtained by equating the corresponding coefficients on the two sides of Eq. (1.2). The recursion is as follows: ⎧ p ⎪ d0,n,k = dpm,0,k = dp0,0,k = 0; ⎪ ⎪ ⎨ 2n−2 i m−3 (2.1) p p p p−1 ⎪ ⎪ = d + d + k d , d ⎪ i+1 ˙ ˜ m,n,k s,n−1,k ⎩ m+k −2,n−1,k m−k −2,n−1,k s=m−1
j=1
j
ki+1 ≥1
i+1
A Census of Petal Bundles by Genus
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where m, n, k ≥ 1. Theorem 2.1 recursion :
The number of rooted petal bundles on Op (p ≥ 1) satisfies the following Gpm,n =
2n−2 s=m−1
Gps,n−1 +
m−3 k1 ≥1
k1 dp−1 m−k1 −2,n−1,k1 .
(2.2)
Proof It is easy to get the above result from formula (2.1). For example, when i = 0, p = 1 for Eq. (1.2), the following theorem can be obtained by using Lagrangian inversion on its parametric expression. Let n−m−p (i + q)!λ(i) 2n − m − 1
. A(p, q, λ) = q!i! n−m−p−i
(2.3)
i≥0
Theorem 2.2
The enumerating function g1 (x, y) has the following explicit expression : G1m,n xm y n , g1 (x, y) = m,n≥1
where
G14,2
= 1, and m! A(1, 2, 1) 2(m − 2)! m! m! + A(0, 1, 1) + A(−2, 0, 1). 3!(m − 3)! 4!(m − 4)!
G1m,n = mA(2, 3, 1) +
Corollary 2.1 [12]
The number of rooted petal bundles on the torus with size n is : n−2 i≥0
3
(2.4)
(2n)!(i + 3)! . 6i!(n + i + 2)!(n − i − 2)!
Nonorientable Rooted Petal Bundles
For Liq (i ≥ 0, q ≥ 1), it can be separated into four categories: LiqI , LiqII , LiqIII and LiqIV : • M ∈ LiqI , er (M ) is a single edge and the j-th (j = 1, 2, . . . , i) distinguished non-rooted face is adjacent to the root-edge; • M ∈ LiqII , er (M ) is a single edge and any distinguished non-rooted face is not adjacent to the root-edge; • M ∈ LiqIII , er (M ) is a double edge and M − er (M ) is still nonorientable; • M ∈ LiqIV , er (M ) is a double edge and M − er (M ) becomes orientable. Similarly to Lemmas 2.1 and 2.2, the following two lemmas can be derived: q
q
Lemma 3.1
q . Let Li I = {M − er (M ) | M ∈ LiqI }. Then Li I = Li−1
Lemma 3.2
Let Li II = {M − er (M ) | M ∈ LiqII }. Then Li II = Liq .
q
q
For any map on a surface, it is easy to see after deleting a double edge the number of the face can be increased by not more than one. Thus it can be seen that the genus of the nonorientable surface will be reduced by two at most after doing that. Therefore, LiqIII can q q be further divided into two parts: Li III1 = {M ∈ LiqIII | M − er (M ) ∈ Liq−1 } and Li III2 = q−2 } {M ∈ LiqIII | M − er (M ) ∈ Li+1
Xu Y. and Liu Y. P.
942 qIII1
Let Li
Lemma 3.3
qIII1
q
= {M − er (M )|M ∈ Li III1 }. Then Li qIII
= Liq−1 .
Proof It is obvious that Li 1 ⊆ Liq−1 . On the other hand, for any M ∈ Liq−1 , there are q m(M ) + 1 ways to obtain M ∈ Li III1 by adding a new double edge from the tail of er (M ) to any vertex on the root-face boundary of M as the root-edge of M . It is evident that there are (m + 1) places where the head of the new root-edge can be attached. Since M = M − er (M ), qIII1
then Li
⊇ Liq−1 . This lemma is true. qIII2
Let Li
Lemma 3.4
qIII2
q
= {M − er (M )|M ∈ Li III2 }. Then Li
q−2 = Li+1 .
Proof Using the same method as for Lemma 2.3, we can obtain the above result easily. q
As for LiqIV , there are two circumstances: Li IV1 = {M ∈ LiqIV | q is odd number} and q Li IV2 = {M ∈ LiqIV | q is even number}. qIV1
Let Li
Lemma 3.5
qIV1
q
= {M − er (M )|M ∈ Li IV1 }. Then Li q Li IV1 ,
q−1 2
= Di
.
q Li IV1 ,
according to the definition of it can be seen that Proof For any map M ∈ Eul(M ) (i.e. v(M ) − (M ) + φ(M ) = 2 − q) is an odd number. However, M = M − er (M ) is an orientable map, so the Eul(M ) (i.e. v(M ) − (M ) + φ(M ) = 2 − 2p) is an even number. Since v(M ) = v(M ) and (M ) = (M ) − 1, then φ(M ) and φ(M ) have the same parity. On the basis of the above depiction, we have φ(M ) = φ(M ). Furthermore, p = q−1 2
M ∈ Di
qIV1
. Li
q−1 2
⊆ Di
q−1 2 .
Therefore,
. q−1 2
Conversely, for any M ∈ Di
, since the Euler characteristic of M (v(M ) − (M ) + φ(M ) = 3 − q) is an even number, it easy to see that q is an odd number. Then, as in Lemma 3.3, a new map M can be obtained by adding a new edge which must get across a cross cap. We have v(M ) = v(M ), (M ) = (M ) + 1 and φ(M ) = φ(M ). Accordingly, v(M ) − (M ) + φ(M ) = q−1 qIV qIV q 2 − q. Furthermore, M ∈ Li IV1 and M ∈ Li 1 , so Li 1 ⊇ Di 2 . This lemma is correct. Lemma 3.6
qIV2
Let Li
q
qIV2
= {M − er (M )|M ∈ Li IV2 }, then Li
q
−1
2 = Di+1 .
Proof From all the above discussion, the lemma can be proved easily. Let Lqi s (s = 1, 2, 3, 4) be, respectively, the enumerating functions of LiqI , LiqII , LiqIII and q q LiqIV ; Lqi 3t and Lqi 4t be the enumerating functions of Li IIIt and Li IVt (t = 1, 2), respectively. Then as a direct consequence of Lemmas 3.1–3.6, the following formulas hold as long as we q q−2 , there are 2ki+1 ways to get M ∈ Li III2 : observe that, for any M ∈ Li+1 i xyzj [zj Lqi−1 (zj , y, z) ˙ − xLqi−1 (x, y, z)], ˙ z − x j j=1 ∗ xy [Lqi − xLqi ], =y (x + x2 + · · · + xm+1 )y n z k = 1−x q
Lqi 1 = Lqi 2
M ∈Li
∗
where Lqi = Lqi (1, y, z); and, Lqi 31 = y
(m + 1)xm+2 y n z k = x2 y(xLq−1 ) , i
M ∈Liq−1
Lqi 32 = y
q−2 M ∈Li+1
2ki+1 xm+ki+1 +2 y n z k = 2x3 y
∂Lq−2 i+1 (x, y, z, x), ∂zi+1
A Census of Petal Bundles by Genus
Lqi 41 = y
943 q−1 2
(m + 1)xm+2 y n z k = x2 y(xDi
) ,
q−1 2
M ∈Di
Lqi 42
q
=y
ki+1 x
q
−1
2 ∂Di+1 y z =x y (x, y, z, x). ∂zi+1
m+ki+1 +2 n k
−1
2 M ∈Di+1
3
Therefore, the theorem B holds. Similarly, from Eq. (1.3), the following recursion can be obtained: q = lm,n,k
2n−2 s=m−1
q ls,n−1,k +
i j=1
q lm+k
˙
j −2,n−1,k q−1
q−1 2 + (m − 1)[lm−2,n−1,k + γq dm−2,n−1,k ] m−3
+
ki+1 ≥1
q−2 ki+1 [2lm−k
q
˜ i+1 −2,n−1,k
−1
2 + γq+1 dm−k
˜ ],
i+1 −2,n−1,k
(3.1)
where m, n, k ≥ 1. Theorem 3.1 recursion :
The number of rooted petal bundles on Nq (q ≥ 1) satisfies the following q Fm,n
2n−2
=
s=m−1
+
m−3 k1 ≥1
q−1
q q−1 2 Fs,n−1 + (m − 1)[Fm−2,n−1 + γq Gm−2,n−1 ] q
−1
q−2 2 k1 [2lm−k + γq+1 dm−k ]. 1 −2,n−1,k1 1 −2,n−1,k1
(3.2)
Let i = 0 and q = 1, 2, 3 for Eq. (1.3). Some known results (which were shown in literature [12]) can be obtained. We list them as follows: The enumerating function f1 (x, y) has the following explicit expression : 1 f1 (x, y) = Fm,n xm y n ,
Theorem 3.2 [12]
m,n≥1 1 1 = 1, F3,2 = 3 and where F2,1 1 Fm,n =m
k≥0
Corollary 3.1
2n − m − 1 m(m − 1) 2n − m − 1 . + 2 n−2 k
n−m−1
The number of rooted petal bundles on the projective plane with size n is 22n−1 −
(2n)! . 2(n!)2
The enumerating function of f2 (x, y) has the following expression : 2 f2 (x, y) = Fm,n xm y n ,
Theorem 3.3 [12]
m,n≥1
where
2 F4,2
= 4,
(3.3)
2 F3,2
= 0,
2 F5,3
2 Fm,n =
= 20 and
n−m−2 i≥0
n−m−1 m(m − 1)σi−1 (m, n) mσi (m, n) + 3i! 2i! i≥0
Xu Y. and Liu Y. P.
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+
m! m! A(0, 1, 1) + A(−2, 0, 1), (m − 3)! 6(m − 4)!
where σi (m, n) = Corollary 3.2
(3.4)
(i + 3)!(3n − m + i + 2) 2n − m . 2n − m n−i+2
The number of rooted petal bundles on the Klein bottle with size n is n−2 i≥0
Theorem 3.4 [12]
(2n)!(3n + i + 4)(i + 3)! . 3i!(n + i + 3)!(n − i − 2)!
The enumerating function of f3 (x, y) has the following expression : 3 Fm,n xm y n , f3 (x, y) = m,n≥1
3 3 = 41, F7,4 = 287 and where F6,3
m m A(4, 5, λ1 ) + A(3, 4, λ2 ) 6 30 m(m − 1) m! A(2, 4, λ3 ) + A(1, 3, λ4 ) + 10 4!(m − 3)! 85m! 62m! A(0, 3, 1) + A(−1, 2, 1) + 4!(m − 4)! 5!(m − 5)! 41m! + A(−3, 1, 1), 6!(m − 6)!
3 = 12mA(5, 6, 1) + Fm,n
where
(3.5)
⎧ ⎪ λ1 (i) = 75i + 36m + 414; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ λ2 (i) = 41i2 + 40m2 + 225mi + 1005m + 226i + 185; ⎪ ⎪ λ3 (i) = 41i + 115m − 25; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ4 (i) = 51i + 52m + 48.
Corollary 3.3
The number of rooted petal bundles on N3 with size n is n−5
(n + 2)!(82n2 − 130n − 102) (i + 6)!(2n)!σi (n) + , 6!(n − 3)! 6!i!(n + i + 5)!(n − i − 3)! i≥0
2
2
where σi (n) = 128n − 22i + 58ni + 435n − 147i − 161. Now let i = 0, q = 4 for Eq. (1.3). A series of equations and parametric expressions concerning the enumerating function f4 (x, y) can be derived. Employing Lagrangian inversion on its parametric expression the following results can be obtained: Theorem 3.5
The enumerating function of f4 (x, y) has the following expression : 4 Fm,n xm y n , f4 (x, y) = m,n≥1
4 = 488 and where F8,4 4 = 120mA(7, 9, 1) + Fm,n
4m m A(6, 8, λ1 ) + A(5, 7, λ2 ) 3 18
A Census of Petal Bundles by Genus
945
m m(m − 1) A(4, 6, λ3 ) + A(3, 6, λ4 ) 504 336 m! 5m! A(2, 5, λ5 ) + A(1, 5, λ6 ) + 252(m − 3)! 6!(m − 4)! m! 984m! A(0, 4, λ7 ) + A(−1, 4, 1) + 600(m − 5)! 6!(m − 6)! 1330m! 488m! + A(−2, 3, 1) + A(−4, 2, 1), 7!(m − 7)! 8!(m − 8)! +
(3.6)
where ⎧ ⎪ λ1 (i) = 106i + 45m + 909; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ2 (i) = 475i2 + 288m2 + 1431im + 6644i + 10584m + 23328; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ3 (i) = (m − 1)(m − 2)(13752i + 2m + 96258) + (i + 7)(i + 8)(488i + 8550m − 4158); ⎪ ⎪ ⎪ ⎨ λ4 (i) = 1464(i + 7)(i + 8) + 15744(i + 7)(m − 2) + 12208(m − 2)(m − 3); ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ5 (i) = 570(i + 6)(i + 7) + 3409(m − 3)(i + 6) + 966(m − 4)(m − 3); ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ6 (i) = 656i + 2100m − 4464; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ7 (i) = 800i + 1045m − 1225. Corollary 3.4
The number of rooted petal bundles on N4 with size n is (n + 3)!(1760n2 − 7056n + 35200) (n − 4)!9! +
n−7 i≥0
+
n−5 i≥0
(i + 9)!(2n)!(1392n + 1152i + 8184) i!9!(n + i + 7)!(n − i − 6)! (i + 9)!(2n)!(2388n − 1412i − 5160) . i!9!(n + i + 5)!(n − i − 4)!
∞ q q Let F.,n (x) = m=1 Fm,n xm . According to formula (3.6), it 4 (x) = 488x8 ; • F.,4 4 (x) = 488x + 488x2 + 570x3 + 656x4 + 800x5 + 984x6 • F.,5
is easy to check : + 1330x7 + 1952x8 + 4392x9 .
References [1] [2] [3] [4] [5] [6]
Liu, Y. P.: Enumerative Theory of Maps, Kluwer, Dordrecht/Boston/London, 1999 Liu, Y. P.: Theory of Counting Combinatorial Maps (in chinese), Science Press, Beijing, 2001 Tutte, W. T.: A Census of Planar Maps. Canad. J. Math., 15, 249–271 (1963) Tutte, W. T.: On the enumeration of planar maps. Bull. Amer. Math. Soc., 74, 64–74 (1968) Brown, W. G.: Enumeration of nonseparable planar maps. Canad. J. Math., 15, 526–545 (1963) Bender, E. A., Canfield, E. R., Robinson, R. W.: The enumeration of maps on the torus and the projective plane. Canad. Math. Bull., 31, 257–271 (1988) [7] Liu, Y. P.: Advance in Combinatorial Maps (in Chinese), Northern Jiaotong University Press, Beijing, 2003 [8] Cai J. L., Liu, Y. P.: The enumeration of general rooted planar maps. Acta Mathematica Sinica, English Series, 21(1), 215–224 (2005) [9] Ren, H., Liu, Y. P: Enumeration of 4-regular maps on the sphere and on the Torus. Discrete Math., 15, 273–288 (2001)
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[10] Li, Z. X., Liu, Y. P.: Chromatic sums of singular maps on some surfaces. J. Appl. Math. Comput., 15, 159–172 (2004) [11] Meir, A., Moon, J. W.: Survival under random coverings of trees. Graphs and Combinatorics, 4, 49–56 (1988) [12] Xu, Y., Liu, Y. P.: Counting pan-fan maps on nonorientable surfaces. Ars Comb., 83, 15–32 (2007)
