$P$-partitions and $p$-positivity

TL;DR

This paper advances the theory of quasisymmetric functions by deriving new expansion formulas and positivity results for generating functions of reverse P-partitions, impacting various combinatorial polynomials and functions.

Contribution

It introduces two new results: the expansion of the fundamental basis into quasisymmetric power sums and the positive expansion of reverse P-partition generating functions, with broad applications.

Findings

Fundamental basis expands into quasisymmetric power sums

Generating functions of reverse P-partitions expand positively into quasisymmetric power sums

Nonnegative linear combinations of these functions are p-positive when symmetric

Abstract

Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.