Enumeration of m-ary cacti

Details Enumeration of m-ary cacti Enumeration of m-ary cacti

1998-04-24

arxiv.org/abs/math/9804119v2

Advances in Applied Mathematics, 24 (2000), 22-56

Mathematics

Combinatorics

LaTeX2e, 28 pages, 9 figures (eps), 3 tables

Scientific paper

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 labelled and unlabelled m-ary cacti, according to i) the number of polygons, 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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