Counting Rooted Non-separable Nearly Cubic Planar Maps *

Cai J unliang

Laboratory of Mathematics and Complex Systems

School of Mathematical Sciences, Beijing Normal University

Beijing, 100875, China. caijunlinag@bnu.edu.cn

Abstract

In this paper, the enumerating problem of rooted non-separable nearly

cubic planar maps with the valency of the root-vertex and, the size and

valency of the root-face of the maps as three parameters will been discussed.

A new method used, and simpler results can been derived in this paper.

Key Words : planar map, non-separable map, Lagrangian inversion.

MR (1991) : Subject Classification 05C45

§1 Introduction

The enumeration of rooted planar maps was originally discussed by W. T. Tutte in the 1960's [13]. Since then, much work has been performed

by numerous scholars including E. A. Bender [1], W. G. Brown [2,3], Z. C.

Gao [5,6], I. J. Good. [7], Y. P. Liu [8 11], R. C. Mullin [12] and W. T.

Tutte himself [13,14].

A planar map is a 2-cell imbedding of a connected graph, loops and

multiple edges allowed, on the sphere. The size of a planar map is the

number of its edges. A planar map is rooted if an edge is distinguished

as the root-edge and half of the root-edge is distinguished from the other

half as the root. The vertex incident to the root is called the root-verte:1;

and the face on the right of an observer on the root-edge facing away from

the root is called the root-face or the outer face. The root, root-vertex,

*Project 11371133 Supported by NNSFC.

ARS COMBINATORIA 137(2018), pp. 53-70

root-edge and root-face of a map all are called root-elements of the map.

The other vertices of the map are called non-root vertices of the map. The

other edges and faces of the map are called the inner edges and inner faces,

respectively, of the map.

Two rooted planar maps are considered combinatorially equivalent if a

homeomorphism of the plane exists, which transforms one into the other,

preserving the root-elements.

A map M is said to be separable if its edge-set E = E(M) can be

partitioned into two disjoint non-null subsets S and T, i.e. E = SU T and Sn T = 0, so that just one vertex v is incident with both members

of S and members of T, i.e. M[S] n M[T] = { v }. The vertex v is called

a separable-vertex of the map. A map without a separable-vertex is called

non-separable.

A vertex v of a planar map M is said to be a cut-vertex of the map

NJ if the map M - v has at least two non-null connected component. A

map without a cut-vertex is called 2-connected. Of couse, a cut-vertex of a

planar map must be a separable-vertex of the map. But a separable-vertex

of a planar map is not necessarily a cut-vertex of the map.

A map with only 3-valent vertices is called cubic map. A rooted map

all of vertices except possibly the root-vertex are 3-valent is called a nearly

cubic map.

The dual of a planar map M, denoted M*, is a planar map obtained

by placing a vertex of M* in each face of M and an edge of M* across

each edge of M. The dual of a cubic planar map is a triangulation which

is played an important role in solving the Gaussian crossing problem and

four color problem [12,13].

As a researcher of counting problems, we prefer to consider the more

general problem of counting near triangulations (rooted planar maps all of

whose faces except possibly the root-face are 3-valent) with several param-

eters. To solve this problem, we found it more convenient to pass through

the dual problem of counting nearly cubic planar maps.

Therefore, we consider the problem of counting how many combinatori-

ally inequivalent, rooted, and non-separable nearly cubic planar maps exist

54

with the valency of the root-vertex and, the size and the valency of the

root-face of the maps as given parameters. Although this issue has been

studied previously, the results are not simple, and the previous study was

not thorough. We state and discuss the question completely. In this paper,

a new method will be used, and some simple results will also be derived.

Terminologies and notations not explained here refer to [7,10,15].

For any rooted map M, let er(M) be the root edge of M, and let

M • er(M) be the result of the rooted map after contraction of er(M)

from M with the successive edge of the root-edge on the boundary of the

root- face as the new root-edge .

Let Nnc be the set of all rooted non-separable nearly cubic planar maps,

and NJ~) be the set of all rooted non-separable cubic planar maps. Clearly,

NJ~)~ Nnc•

For any set M Nnc as above we define an enumerating function as

follows: fM(x,y,z) = L xm(M)yn(M)zl(M),

MEM

where m(M), n(M) and l(M) are the valency of the root-vertex and, the

size and the valency of the root-face of M E M, respectively.

Further, let fnc = fN"Jx,y,z),

and

{ hnc = hN,,_Jx,y) = fNnJx,y, 1) = L xm(M)yn(M);

MENnc

Fnc = FN,.Jy,z) = fN,.Jl,y,z) = L yn(M)zl(M), MENnc

{ Fnc = FN,.Jy,z) = JN,.Jl,y,z);

Hnc = HN,,JY) = hNnJl, y) = FNnJY, 1) = !Nnc(l, Y, 1),

H;,!l = H N~!l (y), F~~) = F N~!l (y, 1).

§ 2 Parametric Expressions (I) for Nnc

The enumerating problem of rooted nonseparable nearly cubic planar

maps was discussed [4] in 1999, but some results and methods used in that

time were not satisfactory. The results are referenced as follows.

55

Lemma 2.1 [4] The generating function fnc = J N,,Jx, y, z) for enu-

merating rooted non-separable nearly cubic planar maps with three param-

eters satisfies the following functional equation :

(x - yz - yzhnc)fnc = x 2yz(x - yz - yzFJ~l). (1)

Lemma 2.2 [4] The generating function Hf;,) = H N,;!l (y) for enumer-

ating rooted non-separable cubic planar maps has the following parametric

expressions :

(2)

Lemma 2.3 [4] The generating function hnc = hN,,Jx, y) for enu-

merating rooted non-separable nearly cubic planar maps has the following

parametric expressions :

{ xy 2 = ((1 - ()(l - rJ)2 , 2y 3 = rJ(l - r7)2 ;

2y 3 hnc = e(l -T)) 2 [2(1 - 0(1 - TJ) - TJ]. (3)

Lemma 2.4 [4] The generating function Fr~~) = F N,\;,l (y, z) for env,mer-

ating rooted non-separable cubic planar maps has the following parametric

expressions :

{

3 2 2( 2y =rJ(l-rJ), Z= l-(l- 2()(l+()TJ;

y 3 z(l + F,~~)) = TJ((l - rJ - TJ(). (4)

Lemma 2.5 [4] The generating function fnc = JN,,.(x, y, z) for enu-

merating rooted non-separable nearly cubic planar maps has the following

parametric expressions :

l xy 2 = ((l -()(l -TJ) 2 , 2y 3 = rJ(l - rJ)2 ,

-1 -2 2( X y z = (l - ()(l - T))2[1 - (1 - 2(()(1 + (()TJ];

-2 2 (1 - l)(l - T/)2 - (1 - T/ - (TJ()TJ( X Y fnc = (TJ( (1 - ()(l - 2(2()(1 - T/ - TJ() .

(5)

56

In the paper [4], the explicit expressions of the enumerating functions

H,\~) = H N,\}l (y), hnc = h,v-,,,(x, y) and FJ~l = F N~;l (y, z) were derived

in 1999 by employing Lagrangian inversions with one [7] or two variables

[15] from (2), (3) and (4) as above, respectively, and in which two of

them are sum-free formulas. However, for the enumerating function fnc =

f,v-,,,(x,y,z) of (5) as well as Hnc = H,v-,,,(y) and Fnc = F,v-,,,(y,z) none

of the results were calculated in that paper because of the obvious com-

plications involved. Now, we perform these calculations by simplifying the

parametric expressions (5) that were first used in that paper.

§3 Parametric Expressions (II) for Nnc

To simplify the parametric expression (5), we have to simplify the para-

metric expressions (2), (3) and (4) first.

Lemma 3.1 The generating functions of Hi~) = H N\,~l (y) and hnc = h,v-,,, (x, y) for enumerating rooted non-separable nearly cubic planar maps

have the following parametric expressions :

{ xy- 1 = a(l - a/3),

H(3l = /3(1 - 4/3) nc (1 _ 2{3)2 '

y3 = /3(1 - 2/3)2;

hnc = {3a2 [ (1 - 2/3) (1 - a/3) - /3]. (6)

Proof By substituting T/ = 2(3 into the formulas of (2) and (3) we can

find that

:cy2 = ~(l-O(l-r1) 2 = ~(1-~)(1-2/3) 2, 2y3 = TJ(l-TJ)2 = 2/3(1-2/3) 2,

{ 4y 3 Hi~l = 'f/2(1 - 2TJ) = 4{32(1 - 4/3),

2y 3hnc = e(1 - 2/3)2[2(1 - ~)(1 - 2/3) - 2/3].

That is y 3 = /3(1- 2/3)2, Hi~)= /32(1 4{3)y- 3 = ~;1_-2;1, and

xy 2 = ~(1 - ~)(1 - 2(3)2 , y3hnc = e(l - 2/3)2[(1 - 0(1 - 2/3) - /3].

Now letting~= a/3, the above becomes

xy- 1 = a(l - a/3), hnc = {3a2[(1 - a/3)(1 - 2/3) - /3].

57

The lemma holds .

Lemma 3.2 The generating function FJ~) = F N~~,> (y, z) for enumer-

ating rooted non-separable cubic planar maps has the f ollowi,ng parametric

expression :

{ y3 = ,8(1 - 2,8)2' z = 1 + ,B-y + (~ - 2,B),B-y2;

FJ~) = ,B-y2[(1 - 2,8)(1 - ,B-y) - ,B]. (7)

Proof From (1) we can find that if x - yz - yzhnc = 0, then it must

also be x - yz - yzF 1~~) = 0. That is

xy-l z - FJ~) = xy-lz- 1 - l.

- 1 + hnc'

Then we can substitute (6) into the above equations, and after grouping

the terms, we have

o:(1 - ,Bo:) 0:

1 + ,Bo:2[(1 - ,Bo:)(1 - 2,8) - ,B] 1 +,Bo:+ (1 - 2,8),Bo:2 '

and FJ~) = xy- 1z- 1 - 1 = (1 - ,Bo:)[1 +,Bo:+ ,Bo:2 - 2,B2o:2] -1. Thus

FJ~l = ,Bo:2[(1 - 2,8)(1 - o:,B) - ,B].

By substituting a new parameter 'Y for o: in the formulas above, the

lemma holds immediately.

Lemma 3.3 The generating function fnc = f NnJx, y, z) for enumer-

ating rooted non-separable nearly cubic planar maps has the following para-

metric expression :

{ xy- 1 = o:(1 - o:,B), y3 = ,8(1 - 2,8)2,

'Y f 2 o:-y(l - o:,B - ,B-y) (8) z = 1 + ,B-y + (1 - 2,B),B-y2; nc = xy 1 - (1 - 2,B)o:,B-y ·

Proof From (1) of Lemma 2.1, we have

-1 (1 r;,(3)) f _ 2 xy - Z + I'nc nc - X yz xy-l - z(l + hnc) ·

58

We can substitute (6) and (7) into it as above and grouping the terms,

we have

xy- 1 = a(l - a/3), y3 = /3(1 - 2/3)2 , z= ' 1 + /3, + (1 - 2(3)(3,2'

and

2 a(l - a/3) - H/3,+<f- 2/3)/3,2 {1 + /3, 2[(1 - 2/3)(1 - /3,) - /3]} fnc = x yz a(l - a/3) - H/3,+(f- 2/3)/3,2 {1 + /Ja2[(1 - 2/3)(1 - f3a) - /3]}

2 (1 - a/3 - ,(3), 2 a,(1 - a/3 - 1/3) =xy =xy .

(1 - a/3)[1 - (1 - 2/J)a/J,] 1 - (1 - 2/J)a/3,

This is the result.

Lemma 3.4 The generating function Hnc = HNnJY) for enumerating

rooted non-separable nearly cubic planar maps has the fallowing parametric

expression :

0 y= 1+20 3'

Proof Because Hnc = HN,,JY) = hNnJl, y), from (6), we have

(9)

y- 1 = a(l - a/3), y3 = /3(1 - 2(3)2; Hnc = (3a2 [(1 - 2/3)(1 - a/3) - /3].

Letting f3 = 03 /(1 + 203 ) , we then have

0 /3 1 /3 y = l + 203 = 02 = a(l - a/3) = a(J(l - a/3)'

i.e., a/3(1 - a/3) = 02 .

Further, letting a/3 = >.0, we have >.0(1- >.0) = 02 , i.e. 0 = >./(1 + >.2 ).

Thus 03 >. >. 3

/3 = 1 + 203, 0 = l + )-.2' a= 02 (1 + 20 ).

The following parametric expression can then be checked

Hnc = (3a2 [(1 - 2/3)(1 - a/3) - /3] = a>.0[(1 - 2/3)(1 - >.0) - /3]

= >.0[(a - 2>.0)(1 - >.0) - >.0] = >.20- 1[1 - >.0 - 03]

[ >.3 ] 2 2 = >. 1 - (1 + )-.2)2 = >. - >. 0 .

59

This is the result.

Lemma 3.5 The generating Junction Fnc = FNnc (y, z) for enumerating

rooted non-separable nearly cubic planar maps has the following parametric

expression :

0 y = l + 203 '

z - 'Y . - 1 + /h + (1 - 2(3)(3"(2 '

1 - .\0"fy Fnc = 'YY l , ,

-A"(Y

03 = 02 0 = A 0' = l - .\ 2 where (3 = l + 203 Y, 1 + ,\2 · >- (l + _\2)2 ·

(10)

Proof Because Fnc = FN,,JY, z) = fN,,,(l, y, z), from (8), we have

{ y~ 1 = a(l - a(3), y 3 = (3(1 - 2(3)2, z = l + (3"( + (~ _ 2(3)(3"(2;

Fnc = Y2 CX"f(l - a(3 - (3"(). 1 - (1 - 2(3)af3'Y

Therefore, following from the proof of Lemma 3.4, we can then let

g3 (3 = l + 203 ' a(3 = .\0' 0=->--

l + _\2'

0 y = l + 203 .

Thus, we have

2 a"((l - a(3 - (3"() 02a"((l - a(3 - (3"() = y l - (1 - 2(3)a(3"( = (1 + 203 ) 2[1 - (1 - 2(3)af3'Y]

A'f[l - (1 + .\2)f3'Y] B'Y[l - (1 + .\2)f3'Y] 1 - .\0"fy (1 + .\2)(1 + 203 ) - .\ 2"/ = 1 + 203 - 0.\"f = 'YY l - A"fY .

This is the result.

§4 Explicit Expressions for Enumerating Nr,c

The purpose of this section is to discover the explicit expressions for

the enumerating functions inc = f N,,, ( x, y, z), H nc = f Nn, ( 1, y, l) and

F,,c = fN,,, ( 1, y, z) by using Lagrange inversions .

In fact, from (8) we have the enumerating factor of the generating func-

tions fnc = fN,,Jx, y, z) for enumerating rooted nonseparable nearly cubic

planar maps:

.0.(a,/3,"Y) =

1 - 2a(3 1- a(3

0

*

* 1 - 6(3 1 - 2(3

*

60

0

0

1 - (1 - 2(3)(3"(2

1 + (3"( + (l - 2(3)(3"(2

(1 - 2ap)(l - 6p)[l - (1 - 2p)p,' 2] (1 - ap)(l - 2p)[l +Pi'+ (1 - 2p)p,, 2J ·

(11)

Now, by applying a Lagrangian inversion with three parameters from

(8) and (11), we have

_ 0(m,n,l)a [1 +Pi'+ (1 - 2p)p,' 2]1- 1(1 - ap - Pl') - L.., (c,,(3,-y) ,' (1 - ap)m+ 1(1 - 2p) 2n+1[1 - (1 - 2p)ap,']

-m,ri,l?::._O

fnc

i.e.

fnc == 0(m,n,l) (1 - ap - P!')(l - 2ap)(l - 6p) L.., (a,{3,y) [1 - (1 - 2p)aP,'](1 - ap)m+ 2(1 - 2p) 2n+l

rri,n,/2:_0 (12)

Now, let us discuss some special situations.

Theorem 4.1 The generating function Fl~)= FN,\;>(y,z) of rooted

non-separable nearly cubic planar maps of the valency 2 of the root-vertex

with the size and the valency of the root-face of the maps as two parameters

has the following explicit expression :

Ln,1/2J 2n-l+il!(3n - l - l)!µ;y3n+1 z1+1 F N,<,.2,.> = yz + (13)

L.., L.., i!(i + 2)!(l - 2i)!(n - l + i)!(2n - i + 1)!' n2::l,l2::2 i=O

where µi = [4n + 2 + i(l - n - i -1)](3n - l + 1)(21- 3i) - i(3n - l)(l + 2).

Proof Using m = 0 from (12) we have

n,12::0

[1 - (1 - 2P)P!'2] (1- P!')(l - 6p) Y3n+lzl+l (1 _ 2p)2n+l

rnin(n,l/2) 0 (n-j,l-2j) (1 - P!')(l - 6p)

L.., L.., (13,-y) (1 _ 2p)2n-J+l n,12::0 j=O

[ C) (1 + P!')l-j - C 1) (1 + P!')/-J+l] Y3n+l zl+l.

By spreading out the terms on,', we may have

61

f;\'.' ,~o m]t' a;-'+> { G) [ (1 ; j) ... G D] (1 - 2m

-( l) [(l-j+l)-(l-j+l)] (l-6(3)y 3n+l 2 l+l}· j - l j + l j + 2 (1 - 2(3)2n-y+2

By spreading out the terms on (3, we may also have

min(n,l/2) . { (l) [(l _ ') (l _ ·)] = . Z + " " 2n-z+1 J - J y . . ·+1 n:2:1,1:2:2 j=O J J J

[ ( 3n - l ) ( 3n - l - l ) ] ( l ) [ (l - j + 1) n-l+j - 3 n-l+j-1 - j-1 j+l -

( l j + 1)] [(3n - l + 1) _ 3 ( 3n -_l )] } y3n+lzl+l, J+2 n-l+J n-l+J-1

from which we have the result by grouping the terms. tt Corollary 4.1 The generating function Fr~~) = F N,\~l (y, z) of rooted

non-separable cubic planar maps with the size and the valency of the root-

face of the maps as two parameters has the following explicit expression:

ln,l/2J 2n-l+i[!(3n - l - l)!µiy3nzl

L L i!(i + 2)!(1 - 2i)!(n - l + i)!(2n - i + 1)!' n:2:U:2:2 1=0

FN<"l = /1,('

(14)

wheT'e µi = [4n + 2 + i(l - n - i -1)](3n - l + 1)(21- 3i) - i(3n - l)(l + 2).

Proof Noticing that F NtJ (y, z) = y- 1 z- 1 F N,\~l (y, z) - 1 from (13) we have the result. q

Theorem 4.2 The generating Junction fnc = J N,,, (x, y, z) of rooted

non-separable nearly cubic planar maps with the valency of the root-vertex

and, the size and the valency of the root-face of the maps as three parameters

has the following explicit expression :

p 3p-l-l l-m+l l(1-m-i)/2J+l JN,,, = x2F,(~) + LL L L L 2p-s-l

p:2:2 1=2 m=3 i=O j=O (15) (m + i - 2)!(3p- m - l + 2i - l)!l!mµijXmY 3p-mzl

i!j!r!(l - r - j + 2)!(p - s)!(2p + 2i - r + l)!m!l '

where r = rn + i + j ., s = l - i - j, and

Jlij = (m - ·i - l)[r(r - l)(m + i - 2)(s - p)(2l - r - 2j + 3)+

2(rn + i)(l - r - j + 2)(2p + 2i - r + 1)(2r - l + j - 1)(21 - r - 2j)].

62

Proof Putting l = 0 into (12), we have

fnc I l=O _ a(m,n,O) (1 - a/3 - /31 )(1 - 2a/3) - L.. (a,,6,,) [1 - (1 - 2(3)a(31](1 - a(3)m+2

m,n?':0

[1 _ (1 _ 2/3)/3,2] 1 - 6(3 xm+2y3n-m+l 2 1 (1 - 2(3)2n+l

a(m,n) (1 - 2a(3)(1 - 6(3) m+2 3n-m+l L.. (a,(3) (1 - a(3)m+l(l - 2(3)2n+l X Y Z

rn,n>O

= - [(2m) _ 2 (2m - 1)] an-m (1 _ 6(3)xm+2y3n-m+l z L m m - 1 /3 (1 - 2(3)2n+l m,n2:_0

an 1 - 6(3 2 3n+l = L.,. (3 (1 - 2(3)2n+l X y z

n>O

= I)n [ (3:) _ 3(3:: 11)] x2y3n+1 2 = x2yz, n;::o

and putting n = 0 into (12), we have also

fnc I n=O _ a(m,O,l) (1 - a/3 - (3,)(1 - 2a(3)(1 - 6(3) - . L.. (a,(3,,) [1 - (1 - 2(3)a(31 ](1 - a(3)m+ 2(1 - 2/3)

m,120

[1 + /3, + (1 _ 2/3)/3,2]1[1 _ (1 _ 2(3)(3,2]xm+2yl-m 2 l+l

I: a{:,~?xm+2yl-mzl+l = I: a~x2yzz+1 = x2yz. m,l?".O 120

Thus (12) can be rewritten as

_ 2 (2) (m,n,l) (1 - a(3 - /31 )(1 - 2a(3)(1 - 6(3) .fnc - X Fnc + L.. a(a,(3,,) [1 - (1 - 2/3)af3,](1 - a(3)m+2(1 - 2(3)2n+l

m,n,121

Now, we consider the general situations. By spreading out the terms

on a from (16), we have

lm,nj ( ")I( . ) = 1.2 p(2) = m + i . m - i + 1

fnc " nc L.,. L.,. i!(m + l)! m,n,121 i=O

a(m-i,n-i,l) (1 - a(3 - /3,)(1 - 6(3) . [1 + (3 + (1 _ 2(3)(3 2]1 (a,( 3,,) [1 - (1 - 2(3)af31](1 - 2(3)2n+l 1 1

63

_ L~J (m + i)!(m - i + 1) [a(n-m,l-m-i) (1 - /J'Y)(l - 6/J) - L L i!(m + 1)! ((3,y) (1 _ 2J3)2n-m+i+l

m,n,l~l i=O

(n-m,l-m-i+l) 1 - 6J3 ] [ (l) ( l ) (l /J )] -0(/3,y) (1 _ 2J3)2n-m+i+2 j - j - 1 + 1'

(1 + /J'Y )l-j (1 _ 2J3)1 /Jj 1'2j xm+2y3n-m+l zl+l

Lm,nJ L(l-m-i+l)/2J =X2p(2) + nc L I: I:

2n-l+i+J-l(m + i)!l! i!(m + l)!j!(l - m - i - 2j + 1)! m,n,l~l i=O j=O

(3n - m - l + 2i - l)!µijxm+2y3n-m+lzl+l

(m + i + j + 2)!(n - l + i + j)!(2n - m + i - j + 1)!'

where µij =

(m - i - 1)[2(2l - m - i - 3j)(l - m - i - 2j + 2)(2n - m + i - j + 1)

(m + i)(2m - l + 2i + 3j - 1) - (2l - m - i - 3j + 3)(m + i + j - 1)

(m + i + j)(n - l + i + j)(m + i - 2)].

By regrouping the terms, the theorem holds. q Theorem 4.3 The generating function Hnc = HN,,c (y) of rooted non-

separable nearly cubic planar maps with size as a parameter has the follow-

ing explicit expression:

Ln/2J 2n-2k+ln!(6k - 2n + 1)! n+l HN"Jy) = L L (2k + 2)![(3k - n)!]2(n - 2k)! Y · (l ?)

n~O k=fn/31

Proof By (9) we have

dHnc 4,X3(1 + ,X2)2 - 4,X5(1 + ,X2) 4_x3 = 1 - (1 + ,X2)4 = 1 - (1 + ,X2)3.

Now,by applying a Lagrangian inversion with one parameter from (9),

we have

1 n-1 2)n ( 2.X3 )n ( 4.X3 ) n HN,,JY) = L -:;;_8>-. (1 + .X 1 + (1 + ,X2)3 1 - (1 + ,X2)3 y

n~l

64

[n/3] i [ ( ) ] ( . ) = L ". _2 _ (n + 1) _ 2 n + 1 n - 3i + 1 yn+l n+l i i-1 k-i

n>O ,=o - /!.-'t.=2k-

2i+1n! (2k - 2i + 1)! n+l

(2k + 2)!(k - i)!(k - i)!i! y .

This is the theorem.

The former terms of Hnc = HN,,, (y) are

HN,,,(y) =y+y3+y4+2y5+4y6+9y1 +···

and correspond to the previous graphics as seen in Fig. l

0 0 & 0 .. .. .. .. y 3 y4 2y5 y

Ll a,~@ & .. .. .. .. .. 9y7

/§:) 1W tZJ .. .. .. .. .. Fig. 1 Rooted nonseparab]e nearly cubic planar maps with size :'S: 7.

Theorem 4.4 The generating function Fnc(Y) = FN,,,(Y, z) of rooted

non-separable nearly cubic planar maps with the valency of the root-face and

the size of the maps as two parameters has the following explicit expression:

(1-l)/2n-/-l

FN,,, = L L L Ao,j[SBo(-1) - 4Bo(0) + 4B0 (1) - C0(1)]ynzl n,l::C:l j=D k=O

l(n-1)/2,1-lJ l(1-i-1)/2J n-l+i-1

+ L L L L A;,j[B;(O) - 6B;(-1)+ n,l::C:l i=l j=O

65

7C;(-2) - 6C;(-1) + 7C;(0) - 4C;(l)]ynz 1, (18)

where

A _ _ 2k(l - 2)!(l - j - l)!(n - l + i - l)!>-;,j i,J - k!j!(j + l)!(l - j - l)!(l - i - 2j - l)!(n - l + i - k)!'

B ( ) ( n - l + i + j - 3k + t ) it= ~(n-3l+2i+3j-3k+l)'

C ( ) ( n - l + i + j - 3k + t ) .;t = ~(n-3l+2i+3j-3k)+l'

and A;,j = (n - l + i - 3k)[(l - l)(i + 1) + j(j - 1)].

Proof From (10) we have the enumerating factor of the generating

functions Fnc = F N,.c (y, z) for enumerating rooted non-separable cubic

planar maps:

Fnc

i.e.,

(1 - >-2 )(1 - 6/3) 1 + ,)_2

0

1 - (1 - 2/3)(J,y2

* 1 + /3, + (1 - 2/3)(3,2 (1 - >-2)(1 - 6/3)[1 - (1 - 2/3)/3, 2]

(1 + >-2)[1 + (3, + (1 - 2(3)(3,2] .

(19)

'""" a(n.l) ,(1 + >-2)n- 1(1 + 203 )n[l + (3, + (1 - 2/3)(3,2 ]1- 1 (.-\,,) 1 - >-,y

n,l2'.0

(1- >-2)(1- >-0,y)(l - 6/3)[1- (1- 2{3)(31 2]y"+ 1z1,

Fnc = L 3~- 1(1 + 203 )n- 1(1 + >-2yn-2(1 - >-2 )(1 - 6(3)A(>-)y"z1, (20) n.l2'. 1

where

A(>-) = 31-1 [1 + /3, + (1 - 2/3)/3,2]1-1 (1 - >-0,y)[l - (1 - 2/3)/3,2] ')' 1 - >-,y

= 8~_1 1 ->-0,y'""" [(' 1) _ (l + /3,) (1. - 2)] 1 - >-,y J J - l

J_

= 31-1 l - >-0,y'""" [(l 2) _ /3,(l. - 2)] ' 1 - >-,y J J - l

J_

66

i.e.,

-ee~-1 I: -i>l

j,k'2_o

1-1 (l-i-1)/2 [(l _ 2) ( l _ . _ l ) A(.\) = "°""' "°""' J j l - i - 2j - 1

i=O j=O

-(l. -2) ( l j 1 )] ,\i(l - 2/3)j 131-i-j-lyi J - 1 l - z - 2J - 2

-0 1-1 (l-i-1)/2 [(l -2) ( l - j - 1 ) I: I: j z _ i - 2j - 1 i=l j=O -(l. - 2) ( l j -. 1 )] .\\1 - 2/3)j 131-i-j-lyi

J - 1 l - z - 2J - 2 1-1 (l-i-1)/2

= "°""' "°""' (l - 2)!(l - j - l)![(l - l)(i + 1) + j(j - 1)] f::o j!(l - j - l)!(l - i - 2j - l)!(j + 1)!

,\i(l - 2/3)j 131-i-j-lyi

1-1 (l-i-1)/2 . . .. -0 "°""' "°""' (l - 2)!(l - J - l)![(l - l)(i + 1) + J(J - 1)]

j!(l - j - l)!(l - i - 2j - l)!(j + 1)! i=l J=O

(l-1)/2

that is A(,\)= L ao,j(l - 2/3)j 131-j-l+ j=O

67

l-1 (l-i-1)/2 I: I: ai,j(1 - 0)>-i(1 - 2(3)j (31-i-j- 1yi, (21) i=l j=O

(l - 2)!(l - j - l)![(l - l)(i + 1) + j(j - 1)] where a;j = ---------------,

' j!(l - j - l)!(l - i - 2j - l)!(j + 1)! Thus, from (20) and (21), we have

(1-1)/2 Fnc = L L ao,ja~- 1 (1 + 203)n-l(l + >.2)7'- 2(1 - >.2)

n,/~l j=O

1-1 (l-i-1)/2

+ I: I: I: ai,ja~-i-1(1 + 203)n-1(1 + >-2)n-2(1 - >-2) n,l~l i=l j=O

(1 - 6/3)(1 - 2/3)1 (Jl-i-j-l (1 - 0)yn+i z 1

(1-1)/2

I: I: aa,ja;-3l+3j+2(1 + 203r-l-1(1 - 403) n,l~l j=O

l(n-1)/2,l-lj l(l-i-l)/2J

+ I: I: I: a;,ja;-31+2i+3j+2(1 - 403) n,l~l i=l j=O

68

(l-l)/2n-l-l

F.,c L L L Ao,j[8Bo(-1)-4Bo(0)+4Bo(l)-Co(l)]ynzl n,l;:,:l j=O k=O

l(n-1)/2,l-lj L(l-i-l)/2j n-l+i-l

I: I: I: A-. i,J

i=l j=O k=O

[Bi(O) - 6Bi(-1) + 7Ci(-2) - 6Ci(-1) + 7C;(O) - 4C;(l)]ynz 1

where r; = n - l + i + j - 3k, s; = n - 3l + 2i + 3j - 3k and

A . . _ k [(n - l + i - 1) _ (n - l + i - 1)] .. i,J - 2 k 2 k - 1 a,,1 ,

( r; + t ) B;(t) = ½(s; + 1) ,

By regrouping the terms the theorem holds.

Because the explicit expression of rooted nonseparable nearly cubic

maps with size was never known until now, the answer is provided here.

Corollary 4.2 The generating function Fnc = FNnc (y, z) of rooted non-

separable nearly cubic planar maps with the size and valency of the root-face

of the maps as three parameters has the following explicit expression :

p 3p-l-l 1-m+l l(l-m-i)/2J+l

FN,,, = FJ~) +LL L L L 2p-s-l

p;:,:2 1=2 m=3 i=O j=O

(m + i - 2)!(3p- m - l + 2i - l)!l!mµ;jy 3P-mzl

(22)

i!j!r!(l - r - j + 2)!(p- s)!(2p + 2i - r + l)!m!l'

where r = m + i + j, s = l - i - j, and

µ;j = (m - i - l)[r(r - l)(m + i - 2)(s - p)(2l - r - 2j + 3)

+2(m + i)(l - r - j + 2)(2p + 2i - r + 1)(2r - l + j - 1)(2l - r - 2j)].

Proof Putting x = 1 into (15) we get the result.

69

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