Some combinatorial problems associated with products of conjugacy classes of the symmetric group (Q1824026)

The group algebra of the symmetric group is used to derive a general enumerative result associated with permutations in a designated conjugacy class. For positive integers a, \(t_ 1,...,t_ a\) and a partition \(\psi\) of N, let \(C_{\psi}(t_ 1,...,t_ a)\) be the number of ways of expressing an arbitrary permutation \(\pi\) on N symbols in the conjugacy class indexed by \(\psi\) as a product \(g_ 1...g_ a\) where \(g_ i\) is a permutation on N symbols with \(t_ i\) cycles (in its disjoint cycle decomposition). In this paper are given the generating function \(A_{\psi}(z_ 1,...,z_ a)\) of \(C_{\psi}(t_ 1,...,t_ a)\) and for fixed \(\psi\) and a a number of specializations of this result.