Lecture hall P-partitions

TL;DR

This paper generalizes s-lecture hall partitions to labeled posets, providing new generating function identities and analyzing the real-rootedness and unimodality of related polynomials.

Contribution

It introduces s-lecture hall P-partitions, extends existing identities, and studies polynomial properties for these generalized structures.

Findings

Derived new generating function identities for s-lecture hall P-partitions.

Proved (P,s)-Eulerian polynomials are real-rooted for certain pairs.

Discussed potential unimodality of these polynomials.

Abstract

We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P,s)-Eulerian polynomials are real-rooted for certain pairs (P,s), and speculate on unimodality properties of these polynomials.