Enumerating Sn by associated transpositions and linear extensions of finite posets

Abstract

AbstractWe define a family of statistics over the symmetric group Sn indexed by subsets of the transpositions, and we study the corresponding generating functions. We show that they have many interesting combinatorial properties. In particular we prove that any poset of size n corresponds to a subset of transpositions of Sn, and that the generating function of the corresponding statistic includes partial linear extensions of such a poset. We prove equidistribution results, and we explicitly compute the associated generating functions for several classes of subsets