Poset cone and weighted P−partitions enumerator

Abstract

For a poset P on [n] we associate the poset cone C_P := cone{e_i − e_j : i <_P j}, where e_1, e_2, . . . , e_n are the standard basis vectors in R^n. The normal fan N(C_P) of this polyhedron is a coarsening of a subfan of the braid arrangement fan, given by hyperplanes {x_i = x_j}, for 1≤i<j≤n. Weighted P−partitions enumerator F_q has an algebraic meaning as the universal morphism from a certain combinatorial Hopf algebra of posets P to the Hopf algebra QSym of quasisymmetric functions. For a well labelled poset P and q = 0 enumerator Fq specializes to the classical Gessel's P−partitions enumerator. We also provide an example of posets with the same P−partitions enumerators but which are distinguished by corresponding weighted quasisymmetric enumerators.