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Enumeration of m-Ary Cacti

✍ Scribed by Miklós Bóna; Michel Bousquet; Gilbert Labelle; Pierre Leroux

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
253 KB
Volume
24
Category
Article
ISSN
0196-8858

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✦ Synopsis

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both Ž . labelled and unlabelled m-ary cacti, according to i the number of polygons, Ž .

Ž . ii the vertex-color distribution, iii the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.

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