Inequivalent transitive factorizations into transpositions (Q2762711)

Suppose that \(\sigma=\tau_1\tau_2\cdots\tau_k\) where \(\sigma\) is a permutation of \(N=\{1,2,\dots,n\}\) and \(T=\{\tau_1,\dots,\tau_k\}\) is a set of transpositions which acts transitively on \(N\). This paper deals exclusively with such factorizations of \(\sigma\) which are minimal in the sense that \(k\) is as small as possible. Two factorizations are considered to be equivalent if one can be obtained from the other by a sequence of interchanges of adjacent pairs of commuting transpositions. The main result is a two variable generating function which counts the equivalence classes of factorizations of those permutations which consist of two disjoint cycles. The case of a single cycle had been solved previously, but is rederived in the appendix.