Enumerating $m$colored $m$gonal Plane Cacti
Miklos Bona
Institute for Advanced Study
October 30,
4:15pm
refreshments at 3:45pm
2338
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 wellknown Otter's formula for trees and use
an $m$variable generalization of Chottin's 2variable Lagrange inversion
formula.
\end{abstract}
This is joint work with Michel Bousquet, Gilbert Labelle and Pierre Leroux.

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