## 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 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

Funding Information:This work was partially supported by a grant from the Simons Foundation (#206001 to Margaret Readdy).

Publisher Copyright:

© 2016 Elsevier Inc.

## Keywords

- Algebraic complex
- Discrete Morse theory
- Orthogonality
- Poset
- Stirling numbers
- q-analogues

## ASJC Scopus subject areas

- Applied Mathematics