Department of Mathematics, Statistics and Computer Science, University o~ .... [7] W.T. Tutte, On elementary calculus and the Good formula, J. Combin. Theory ...
235
Discrete Mathematics 54 (1985) 235-237 North-Holland
NOTE
T H E N U M B E R OF L O O P L E S S P L A N A R M A P S Edward A. BENDER* Department of Mathematics, University of Califomia at San Diego, La Jolla, CA 92093, U.S.A.
Nicholas C. WORMALD** Department of Mathematics, Statistics and Computer Science, University o~Newcastle, New South Wales 2308, Australia Received 18 June 1984 We derive a simple formula for the number of rooted loopless planar maps with a given number of edges and a given valency of the root vertex.
Enumeration of rooted planar maps occasionally gives rise to simple formulae for the numbers of maps in non-trivial and interesting cases even when the maps are not trees. Some examples of formulae for rooted planar maps with n edges are:
all (connected) maps (Tutte [5]):
2(2n)! 3" n! (n +2)! '
loopless maps (Walsh and Lehman [8D:
6(4n + 1)! n! (3n+3)! '
3-connected triangulations with n = 3m(Tutte [4]):
2(4m-3)! m! ( 3 m - l ) ! "
See [8] for an explanation of the relationship between the latter two formulae. What about summation-free two parameter formulae for rooted planar maps? Some have been discovered for various sorts of two connected triangulations where one parameter is the number of vertices and the other is the number of vertices on the external face. See Brown [1], Mullin [3] and Tutte [6]. Apparently only one such formula has been discovered so far for other types of planar maps: Brown and Tutte [2] have shown that the number of rooted 2-connected planar * Research partially supported by NSF under grant MCS79-27060. ** Research supported by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowship Scheme. Current address: Depaxlment of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand. 0012-365X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
E.A. Bender, N.C. Worraald
236
maps with i + 1 vertices and ] + 1 faces is
(2i+1-2)!(2]+i-2)t i! i! ( 2 i - 1)! ( 2 ] - 1)! We give another:
For n, m >~1 the number a,,,~ of Ioopless rooted planar maps with n edges and root face valency m is given by
Theorem.
2m(4n-1-2m), (2m + 1) an,,,, =(n-Zm-'~.(~n--_m + 1)t \ m " This is also the number of 3-connected cubic planar maps with 3n + 3 edges and outer face valency m + 2. Proof. By taking dual maps one sees that a,,m is the number of isthmusless planar maps with n edges and with root face of valency m. Let
n=l
m=l
By (5), (10) and (1l) of [9]:
xy2a(x, y)2+ ( 1 - y - xya(x, 1))a(x, y) + y - 1 = 0
(1)
a(~ 1 ) = ( 1 + u)2(1- u)
(2)
1) 4.
(3)
and where u = x(u +
Solving the quadratic equation (1) for a(x, y), using (2) to eliminate a(x, 1) and using (3) to eliminate x, we get
a(x, y)=(l+u)2 ( 2uy 2
y+3uy-(l+u)2±(l+u)(l+u-y)
~/
4uy .~
1 (l+u)2 ].
(4) Calculation of the coefficients of y-2 and y-1 shows that the positive sign, not the negative, is the correct one. By the binomial theorem, ~/
4uy
** 2 ( 2 k 5 : ) (
1 (1 + u) 2= 1 - kY~ ~ =l
uy
~k
(1 +~,W'
and so (4) implies that for k >~ 1 the coefficient of yk in a(x, y) is
1 (2k'~[(k+2)u k u k*l ] (k + 1)(k + 2) ~ k 11(1+ u) 2~-1- (4k + 2) (1~ ~2~ j.
(5)
Applying Lagrange's theorem (see Tutte [7], for example) to extract the coefficient of x" from (5) using (3) establishes the formula for a~,,,. The cubic map result follows from the correspondence in [9]. []
The number o[ loopless planar maps
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References [1] W.G. Brown, Enumeration of triangulations of the disk, Proc. London Math. Soc. 14 (1964) 746-768. [2] W.G. Brown and W.T. Tutte, On the enumeration of rooted non-separable planar maps, Canad. J. Math. 16 (1964) 572-577. [3] R.C. Mullin, Enumeration of rooted triangular maps, Amer. Math. Monthly 71 (1964) 10071010. [4] W.T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962) 21-38. [5] W.T. Tutte, A census of planar maps, Canad. J. Math. 15 (1963) 249-271. [6] W.T. Tutte, The enumerative theory of planar maps, in: J.N. Srivastava et al, exls., A Survey of Combinatorial Theory (North-Holland, Amsterdam, 1973) 437-448. [7] W.T. Tutte, On elementary calculus and the Good formula, J. Combin. Theory Ser. B 18 (1975) 97-137. [8] T.R.S. Walsh and A.B. Lehman, Counting rooted maps by Genus III: Nonseparable maps, J. Combin. Theory Set. B 18 (1975) 222-259. [9] N.C. Wormald, A correspondence for rooted planar maps, Ars Combin. 9 (1980) 11-28.
