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A179299 Coefficients of power series enumerating pentagulations, d-angulations of girth 5. 1
1, 5, 121, 4690, 228065 (list; graph; refs; listen; history; text; internal format)



Bernardi, p.10. A d-angulation is a planar map with faces of degree d. We present for each integer d => 3 a bijection between the class of d-angulations of girth~d and a class of decorated plane trees. Each of the bijections is obtained by specializing a "master bijection" which extends an earlier construction of the first author. Bijections already existed for triangulations ($d=3$) and for quadrangulations (d=4). As a matter of fact, our construction unifies a bijection by Fusy, Poulalhon and Schaeffer for triangulations and a bijection by Schaeffer for quadrangulations.

For d => 5, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate p-gonal d-angulations, that is, d-angulations with a simple boundary of length p. We thereby recover bijectively the results of Brown for p-gonal triangulations and quadrangulations and establish new results for d => 5. A key ingredient in our proofs is a class of orientations characterizing d-angulations of girth d. Earlier results by Schnyder and by De Fraisseyx and Ossona de Mendez showed that triangulations of girth 3 and quadrangulations of girth 4 are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a d-angulation has girth $d$ if and only if the graph obtained by duplicating each edge $d-2$ times admits an orientation having indegree d at each inner vertex.


Table of n, a(n) for n=0..4.

Olivier Bernardi, √Čric Fusy, A bijection for triangulations, quadrangulations, pentagulations, etc, arXiv:1007.1292 [math.CO], 2010-2011.

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

W. G. Brown. Enumeration of quadrangular dissections of the disk, Canad. J. Math., 17 (1965) 302-317.

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.


For d = 5 the series starts as F(x) = x + 5 x^3 + 121 x^5 + 4690 x^7 + 228065 x^9 + ...


Sequence in context: A028448 A108791 A282271 * A012179 A012026 A012190

Adjacent sequences:  A179296 A179297 A179298 * A179300 A179301 A179302




Jonathan Vos Post, Jul 09 2010



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