Enumeration of m-ary cacti according to their color and degree distributions

We enumerate cyclically colored m-gonal plane cacti (called m-ary cacti) according to the degree distribution of vertices of each color. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. The corresponding problem for rooted m-ary cacti has been solved by Goulden and Jackson in connection with factorizations of a cyclic permutation into m permutations with given cycle types. We obtain explicit formulae for both the labeled and unlabeled m-ary cacti, according to both the vertex-degree and vertex-color distributions, and also for unlabeled m-ary cacti according to the order of their automorphism group. To achieve our goal, we prove a dissymmetry theorem for m-cacti extending Otter’s wellknown formula for trees and use anm-variable generalization of Chottin’s 2-variable Lagrange inversion formula.