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arXiv · 1506.03249

q-Stirling numbers: A new view

Yue Cai·Margaret A. Readdy
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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 $Π(n,k)$ 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.

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BibTeXRIS

Yue Cai, Margaret A. Readdy. 2016-11-12. q-Stirling numbers: A new view. https://doi.org/10.1016/j.aam.2016.11.007

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