A $q$-deformation of enriched $P$-partitions (extended abstract)

TL;DR

This paper introduces a unified $q$-deformation framework for classical and enriched $P$-partitions, creating a family of power series that interpolates between known quasisymmetric functions and forms a basis of $ ext{QSym}$ for most $q$ values.

Contribution

It develops a novel $q$-deformation that generalizes previous $P$-partition theories and constructs new bases of quasisymmetric functions parametrized by $q$.

Findings

The $q$-deformed series interpolate between Gessel's and Stembridge's functions.

The series form a basis of $ ext{QSym}$ for $q otin ext{-}1,1$.

New monomial bases extend previous work on enriched monomials.

Abstract

We introduce a $q$-deformation that generalises in a single framework previous works on classical and enriched $P$-partitions. In particular, we build a new family of power series with a parameter $q$ that interpolates between Gessel's fundamental ($q=0$) and Stembridge's peak quasisymmetric functions ($q=1$) and show that it is a basis of $\QSym$ when $q\notin\{-1,1\}$. Furthermore we build their corresponding monomial bases parametrised with $q$ that cover our previous work on enriched monomials and the essential quasisymmetric functions of Hoffman.