# Enumeration of m-ary cacti

- Published in 1998
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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.

## Comment

The number of binary cacti is the number of Truchet tiles.

## Links

### BibTeX entry

@article{Enumerationofmarycacti, title = {Enumeration of m-ary cacti}, 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.}, url = {http://arxiv.org/abs/math/9804119v2 http://arxiv.org/pdf/math/9804119v2}, year = 1998, author = {Miklos Bona and Michel Bousquet and Gilbert Labelle and Pierre Leroux}, comment = {The number of binary cacti is the number of Truchet tiles.}, urldate = {2018-10-23}, archivePrefix = {arXiv}, eprint = {math/9804119}, primaryClass = {math.CO}, collections = {Attention-grabbing titles,Combinatorics,Things to make and do} }