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 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.
Original language | English |
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Pages (from-to) | 50-80 |
Number of pages | 31 |
Journal | Advances in Applied Mathematics |
Volume | 86 |
DOIs | |
State | Published - May 1 2017 |
Bibliographical note
Publisher Copyright:© 2016 Elsevier Inc.
Keywords
- Algebraic complex
- Discrete Morse theory
- Orthogonality
- Poset
- Stirling numbers
- q-analogues
ASJC Scopus subject areas
- Applied Mathematics