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 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 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 kinds are orthogonal.
|Number of pages||31|
|Journal||Advances in Applied Mathematics|
|State||Published - May 1 2017|
Bibliographical noteFunding Information:
This work was partially supported by a grant from the Simons Foundation (#206001 to Margaret Readdy).
© 2016 Elsevier Inc.
- Algebraic complex
- Discrete Morse theory
- Stirling numbers
ASJC Scopus subject areas
- Applied Mathematics