q-Stirling numbers: A new view

We show the classical q-Stirling numbers of the second kind can be expressed more 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 Π(n, k) which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number Sq[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 qStirling 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.