The p- partition generating function and the quasisymmetric basis

An acyclic directed graph can be viewed as a (labelled) poset (P,ω). To the latter, one can associate a (P,ω)-partition generating function which is a quasisymmetric function. We propose two expansions of this function in the recently introduced type-2 quasisymmetric power sums basis φ and derive the leading coefficient of some types of posets