P-partition power sums

TL;DR

This paper extends the theory of P-partitions to weighted labelled posets, introduces new bases for quasisymmetric functions, and provides combinatorial formulas for their algebraic operations.

Contribution

It develops weighted P-partitions, defines related generating functions, and introduces new quasisymmetric power sum bases with combinatorial interpretations.

Findings

New bases for quasisymmetric functions that refine power sum symmetric functions

Formulas for products and coproducts of these bases

Combinatorial interpretations of expansion coefficients

Abstract

We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P-partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases.