# Enumerating $m$-colored $m$-gonal Plane Cacti

## ABSTRACT

We enumerate $m$-colored $m$-gonal plane cacti according to the degree distribution of vertices of each color. We obtain explicit formulae for both the labeled and unlabeled cases. This combinatorial problem is motivated by the classification of complex polynomials having at most $m$ critical values, studied by Zvonkin and others. The corresponding problem for {\em rooted} $m$-cacti has been solved by Goulden and Jackson in connection with factorizations of a cyclic permutation into $m$ permutations with given cycle types. To achieve our goal, we prove a dissymmetry theorem for $m$-cacti extending the well-known Otter's formula for trees and use an $m$-variable generalization of Chottin's 2-variable Lagrange inversion formula. \end{abstract} This is joint work with Michel Bousquet, Gilbert Labelle and Pierre Leroux.