COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE

2000,20B(2):169-174

athemi(c:a9cientia

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f£~~Jm~m COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE 1 Ren Han ( 15:-#)

Liu Yanpei (

:t'Jn~

)

Department of Mathematics, Northern Jiaotong University, Beijing 100044, China

Abstract

It is well known that any kind of exact enumerations of rooted maps on non-

planar surface is quite difficult.

This paper presents a functional equation for rooted

Eulerian maps on the projective plane. A parametric expression, through which an exact enumeration by root-valency and the number of edges of maps may be determined, is obtained.

Key words

(rooted)Map, surface, Lagrangian, inversion, enufunction

1991 MR Subject Classification

1

05C99, 05C10

Introduction A surface is a compact closed 2-manifold. The orientable (non-orientable) surface of genus

k is the sphere with k handles (crosscaps) denoted by Sk(Nk ) . A map M on Sk(Nk ) means that its underlying graph may be drown on (embedded in) it such that no-pair of edges intersect at an inner point and each face is homeomorphic to the disc. A map is rooted if an edge with a direction along the edge, and a side of the edge is distinguished. Two rooted maps are considered to be the same if there is an isomorphism preserving the rooting. A rooted Eulerian map is such a one that all its vertices are evenvalent. In the sixties, Tutte[7] firstly made a break-through in enumeration of planar Eulerian maps.

Later, Brown discussed planar quadrangulations (which are much related to the 4-

regular Illaps[l]). For enumerations of c-nets, Mullin and Schellenberg studied the usage of regular planar Illaps[6]. Since then, Liu investigated in several aspects on the sphere[2-5]. Although a number of closed formulae were found for planar Eulerian maps, there are fewer results for those on non-planar surfaces. To make an approach to count such maps on non-planar surfaces, we follow some ideas in [2] to present SOUle results here. For the surface N1,the projective plane, let Ui and Uo be the sets of rooted Eulerian maps on N 1 and So(the sphere) respectively. Define their enufunctions (i.e., generating functions) respectively as

r, = E MElli

1 Received

x 2m(l\I)ys(Af),

fa

=

E

x 2n t ( l\I

) y s(.1\f ) ,

AIEllo

Feb.l0,1998; revised Apr.15,1999. Supported by NNSFC under Gra.nt No.19831080

170

ACTA MATHEMATICA SCIENTIA

Vo1.20 Ser.B

where 21n(M) and s(M) are the root-valency and the size of M respectively. In [5], the following equation

(1) is found and solved in an explicit formula on x and y . As analyzed later, this will leads to an explicit solution for such maps on the projective plane. Theorem A Let b (1 + 'ltX 2)2(1 - vx 2) be the discriminant of (1). Then

=

F-1

F 1- ( 1, y) - F-1 = 2x"'y JO+ F-1 + / 2 + x ... y 1_ x2 ?

?

(2)

(1 - X2)/2 + x2yFi (1, y) (1 + 'ltx 2 )J 1 - vx 2

(3)

where /2 is the enufunction of such maps in Ui with their root-edge as both a loop and an essential circuit (i.e., noncontractible circuit). Moreover, by using quadratic methods and Lagrangian inversion, we find that

yFi(1,y) =

(X2x~ 1 !2(X,y)) I x

2

(4)

= 1J '

a

2

(5)

/2(X, y) = x y ax (x/o), stands in the case of the projective plane. Theorem B (m ~ 2) edges, is

The number of rooted Eulerian maps on the projective plane with m - 1

. ~ {3 I=2 ( 111.

q=O

2(1n - q - 2) ) m - q - 2 2q + m - q- 2 rri - q - 1

(

2m _) 2m m 13

2} •

To make an exact counting of rooted Eulerian maps on the projective plane, we introduce some relations and find a parametric expression for such enufunction, i.e., Theorem C The enufunction of Ui can be expressed as F iX, ( Y) =

1] - 1 + (1 + (y - 3)1] - 3Y1])x 2 2(1 -1])(1 - x 2 )v 1 - vx 2 1 (2vx - 3x + 1) +2" (1 - x 2)(1 - vx 2 ) 4

2

+

x

2

1 (1 1) = 2" 1 - vx 2 - V1 - vx 2 x 2 ( V1

(1 where

-

vx 2

-

vr=-v)

(

1]X 2

+ ----------;:::=:== 2 2 (1 -1])(1 - x )J 1 - vx

V1 - vx 2 - vr=-vri) ("1- x 2)V 1 - vx 2 2

+

x (J 1 - vx

2

-

J1=V11)

("1- x 2)V1 - vx 2

x 2 )V 1 - vx 2

(1]-1)(1]-2)

y - - - -1]2 ---• -

By a repeated using of Lagrangian inversion one may get an exact enufunction with all coefficients as a summation of all terms positive.

No.2

Ren & Liu: COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE

Theorem D

171

The enufunction of rooted Eulerian maps on the projective plane is

L LL L i+~~

Fi(x,y) ==

k-l~j~ 1

7l-0

1

A(i,j,k,1n,n,s)yi+k-n+s x 2i + 21.:-

2j

71,=0 s~m+n

where

2

Equations for NOll-planar Maps In this section we shall prove Theorem A. We first define an operation on maps,

For two maps M 1 , M 2 with their respective root "i == r(M 1 ) , r2 == r(M2 ) , the map M == M 1 U M 2 , provided M 1 M 2 v with v == v r 1 == v r 2 , is defined to have its root, root-valency, and root-edge are the same as those of M 1, but the root-face is the composition of t- (M1) and

n

=

Ir(M2 ) , where fr(Mi) is the root-face of Mi(i == 1,2) The operation for getting M from M 1 and M 2 is called 1v-production and M is denoted as

Further, for any two sets of maps M

1

and M2' the set of maps

is said to be the 1v-production of M 1 and M

2•

If M == M 1 == M

2,

then we may write

and further,

M0 k ==MOM0 k . The set U 111ay be parted into three parts as

such that

{Mler(M) is a planar loop}, U(i)

==

{

i

== 1,

{Mler(M) is both an essential circuit and a loop}, i == 2, {Mler(M) is a link},

i == 3.

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ACTA MATHEMATICA SCIENTIA

Lemma 2.1

Vo1.20 Ser.B

Let

Then

=Ui0Uo+Uo0Ui, U«:l =Ui'

U~l»

Proof For any map M in uvi, the root-edge e r (M) is a planar loop and its inner and outer domain are different types of maps, aI110ng them one is on the plane and the other a non-planar one. So, u~j> ~ Ui QUo + Uo QUi' On the other hand, for a map M = M 1 0 M 2 in Ui QUo +Uo QUi with M 1 M 2 V r , we can separate M 1 , M 2 by adding a planar loop at V r . Thus, U~lj> is a subset of Ui QUoUo QUi' The first equation holds. Since the shrinking of root-edge (which is a link) in a Eulerian map will lead to another map

n

=

U(:l

in Ui, ~ Ui' Conversely, by the splitting the root-vertex into an edge and its reversibility, we see that for every map M in Ui' there is a 111ap M'(with the root-edge e') in U(3) such that

M = M' • e'. Therefore, Ui

< U<~l.

Let Ii be the enufunction of U(i) for

13 =

~ LJ

= 1,2,3.

Then by Lemma 2.1,

2m(Af)

AfEU i

~ 2i s(~l)_ 2 Fi(1,y)-Fi ' LJ z y - x Y 1 _ x2 i=l

implies (2). After rearranging the items in (2), we have

Notice that the coefficient of Fi is just Vb, where 6 is the discriminant of (1), (3) immediately follows and this finishes the proof of Theorem A. As for the validity of (4), let TJ = x 2 be the double root of b and substitute TJ into the both sides of the above equation, we have

i.e.,(4) stands.

3

Determination of Functions In this section we shall calculate Eulerian maps on the projective plane. Let U 2 b be the set of all the rooted 2-boundary planar maps with their possible odd vertices

on the root-face. Then, one may readily see Lemma 3.1

Let

U'

= U(2b) = {M. er(M) 1M E U2b} ,

U~r> = {M' - e~(M)

1M' E U'}.

Then (i) U(2b) is the 1v-production of the loop map and Uo;

No.2

Hen & Liu: COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE

173

(ii) U~r> = Uo. For a map M in U', there .are 21n(M) - 1 maps in U2b corresponding to M in such a way that

M : x 2m (1\1) yS(AI)

.....-..+

xi t 2nl (k!) - i +2 yS(AI) +1

(1 ~ i

~ 211~( M)

- 1),

where x, t denote the valencies of vertices on the root-face respectively. Thus, 2m( M) just the root-edge valency of the maps corresponding to M. Let t = z , we have

x 2Yf2(x,y,z) =

+ 2 is

L: (21n(M) _1)x2n~(1\f)+2ys(M)+1,

MEU' i.e.,

f 2 (x, Y ) --

2~ (lu/(X,y))

x ox

x

.

Since U~r> = Uo implies fUI (x, y) == x 2yfo(x, y),

h(x, y) = x 2 y :x (xfo(x, y)). Now we begin to calculate Fi(x,y), where Fi(x,y) is the enufunction ofUi. Since 'TJ == x 2 is the double root of fJ,

21J-3+~ y'TJ(1 -

1])

Thus(by( 5)), f2(Vii,Y)==Y'TJlo(~,Y)+

217 - 3 + 1

J* -

3

-1]

.

Substitute Y1]lo(~,Y) == Y21i~~f into Y'TJf2(~'Y) we obtain

1-0 y1]f2(Vii, y) == 0 - 2 + -0-(1 where

__ ('TJ- 1)('TJ- 2)

y -

1]2

y

~

1 - 40),

(}--y1 - 20 .

,-

Therefore,

yFi (1, y) == (} f2( Vii, y) == 02 - 20 + (1 - 0)(1 Employing Lagrangian inversion for
VI - 48). VI - 48) and 0 ==

~, we

have

1{J(8) =

L

ym m-l m! Do=o

m~l

=

{

tp'((}) } (1- 28t'

L: !{3I=2 (2(m- q-2))

m~2 m

q=O

m - q- 2

q

- 2 2q + (2m-3) 2m- 2}ym-l. m - q- 1 m - 1

1n-

From this, Theorem B immediately follows. Next, we begin to prove Theorem C. Since yFi (1, y) == (}12 (~, y), we may rewrite Fi (x, y) as

(6)

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ACTA MATHEMATICA SCIENTIA

Recall that fJ

= (1 + ux 2 )2( 1 -

Vo1.20 Ser.B

vx 2 ) is the discriminant of (1), we have

1 + x 2y - x 2 - (1 + ux 2 )V l - vx 2 fo = 2x"y "(1 - X"")

and further, (2y - 2 - 2uy'1 - vx 2 + (1 + ux 2) v'1~t,X2 )x 2(1 - x 2) h(x, y) = 2(1 _ x 2)2

(1 - 3x 2 )(1 + x 2y - x 2

(1 + ux 2 )V 1 - vx 2 ) 2(1 - x 2)2 -

Since

After simplification, we have

Fi ( x,y )

+ (y - 3)"1 - 3YTJ)x TJX = "I -2(11 +- (1"1)(1 + - x 2 )v l - vx 2 (1 -1])(1 - x2)V1 2

1 (2vx 4 - 3x 2 + 1) +2 (1 - x 2)(1 - vx 2)

=

1(1 2 1-

x

2

(

vx 2 -

+

x

vx 2

vx 2

(1 - x

2)·JI

-

vx 2

x 2( VI - vx 2 - VI - V17) --:-----=---;::======::;:--(1] - x 2)Y!1 - vx 2

1) + VI -

VI -

2

2

(

VI -

vx 2 - vr=-vrJ) (TJ - x 2 )V l - vx 2

vT=V) - vx 2

in which

i.e., Theorem C follows. As a direct conclusion of Theorem C, we may expand Fi (x, y) into power series of z and y to obtain Theorem D. References 1 Brown W G. Enumeration of quadrangular dissection of the disc. Canad J Math, 1965,17: 302-317 2 Liu Yanpei. Enumerative Theory of Maps. Dordrecht, Boston, London: Kluwer, 1999 3 Liu Yanpei. Enumeration of rooted bipartite planar maps (in Chinese). Acta Math Sci, 1989, 9: 21-28 4 Liu Yanpei. 5 Liu Yanpei. 6 Mullin R C, 4: 259-276 7 Tutte W T.

A note on the number of loopless Eulerian planar maps. J Math Res Expos, 1992,12:165-179 On the number of eulerian planar maps. Acta Math Sci, 1992,12: 418-423 Schellenberg P J. The enumeration of c-nets via quadrangulations. J Combin Theory, 1968, A census of slicings, Canad J Math, 1962,14: 708-722