q-Stirling numbers: A new view
arXiv:1506.03249 · doi:10.1016/j.aam.2016.11.007
Abstract
We show the classical -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 and . We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. Its rank generating function is the -Stirling number . 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 -Stirling numbers of the first kind is done. Letting we give a bijective argument showing the -Stirling numbers of the first and second kind are orthogonal.