Enumerating SnSn by associated transpositions and linear extensions of finite posets

We define a family of statistics over the symmetric group SnSn 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 nn corresponds to a subset of transpositions of SnSn, 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.