The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group

Abstract

A combinatorial bijection is given between pairs of permutations in S n the product of which is a given n -cycle and two-coloured plane edge-rooted trees on n edges, when the numbers of cycles in the disjoint cycle representations of the permutations sum to n + 1. Thus the corresponding connection coefficient for the symmetric group is determined by enumerating these trees with respect to appropriate characteristics. This is extended to the case of m -tuples of permutations in S n the product of which is a given n -cycle, in which the combinatorial objects replacing trees are cacti of m -gons.