Positivity among P-partition generating functions

TL;DR

This paper establishes conditions under which the difference of P-partition generating functions for labeled posets is F-positive, extending classical Schur-positivity results to a quasisymmetric setting.

Contribution

It provides necessary and sufficient conditions for F-positivity of differences of P-partition functions and introduces operations that preserve this positivity.

Findings

Identified necessary conditions for F-positivity.

Established sufficient conditions for F-positivity.

Showed that certain poset operations preserve positivity.

Abstract

We seek simple conditions on a pair of labeled posets that determine when the difference of their $(P,\omega)$-partition enumerators is $F$-positive, i.e., positive in Gessel's fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for $F$-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and sufficient.