$P$-partitions and $p$-positivity

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.

arXiv comment47 pages, 4 figures

References in corpus (3)

• Quasisymmetric Schur functions
• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• Order Quasisymmetric Functions Distinguish Rooted Trees

Cited by in corpus (7)

• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• LLT polynomials, elementary symmetric functions and melting lollipops
• Quasisymmetric functions distinguishing trees
• Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions
• Power sum expansions for Kromatic symmetric functions using Lyndon heaps
• Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables
• A weighted Murnaghan-Nakayama rule for $(P,w)$-partitions