Negative q-Stirling numbers

The notion of the negative q-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative q-binomial, we show the classical q-Stirling numbers of the second kind can be expressed 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 the Stirling poset, as well as a homological version of Stembridge’s q = −1 phenomenon. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting t = 1 + q we give a bijective combinatorial argument à la Viennot showing the (q, t)-Stirling numbers of the first and second kind are orthogonal.