Weighted posets and the enriched monomial basis of QSym (extended abstract)

TL;DR

This paper introduces a new basis for the algebra of quasisymmetric functions by extending monomial peak functions to all compositions using weighted posets, generalizing previous results.

Contribution

It develops a weighted poset framework to extend monomial peak functions to all compositions, providing a new basis for QSym and generalizing existing product rules.

Findings

Introduces weighted posets for generating functions.

Extends monomial peak functions to all compositions.

Provides a generalized product rule for the new basis.

Abstract

Gessel's fundamental and Stembridge's peak functions are the generating functions for (enriched) $P$-partitions on labelled chains. They are also the bases of two significant subalgebras of formal power series, respectively the ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao introduced the monomial peak functions, a basis of the algebra of peaks indexed by odd integer compositions whose relation to peak functions mimics the one between the monomial and fundamental bases of QSym. We show that the extension of monomial peaks to any composition is a new basis of QSym and generalise Hsiao's results including the product rule. To this end we introduce a weighted variant of posets and study their generating functions.