q-Stirling numbers: A new view

TL;DR

This paper introduces a new combinatorial and poset-theoretic framework for understanding q-Stirling numbers of both kinds, revealing their polynomial structure, homological properties, and orthogonality through novel decompositions and bijections.

Contribution

It provides a compact expression of q-Stirling numbers using restricted growth words, introduces the Stirling poset of the second kind, and establishes their algebraic and homological properties, including orthogonality with a new parameter.

Findings

q-Stirling numbers expressed as polynomials in q and 1+q

Stirling poset supports an algebraic complex with known homology basis

q,t-Stirling numbers of both kinds are orthogonal

Abstract

We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $\Pi(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done. Letting $t = 1+q$ we give a bijective argument showing the $(q,t)$-Stirling numbers of the first and second kind are orthogonal.