q-Stirling numbers: A new view

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.