Regular Article: Enumeration of m-Ary Cacti

Authors:
Miklós Bóna;Michel Bousquet;Gilbert Labelle;Pierre Leroux
Affiliations:
LACIM22With the partial support of FCAR (Québec) and CRSNG (Canada)., Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada;LACIM22With the partial support of FCAR (Québec) and CRSNG (Canada)., Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada;LACIM22With the partial support of FCAR (Québec) and CRSNG (Canada)., Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada;LACIM22With the partial support of FCAR (Québec) and CRSNG (Canada)., Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada
Venue:
Advances in Applied Mathematics
Year:
2000

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Abstract

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.