{TALK 
        {"October 30}
        {"Enumerating  $m$-colored $m$-gonal Plane Cacti}
        {"Miklos Bona}
        {"Institute for Advanced Study}
        {"}
        {" 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.}
}