## Abstract

The notion of the negative q-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative q-binomial, we show the classical q-Stirling numbers of the second kind can be expressed 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 the Stirling poset, as well as a homological version of Stembridge’s q = −1 phenomenon. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting t = 1 + q we give a bijective combinatorial argument à la Viennot showing the (q, t)-Stirling numbers of the first and second kind are orthogonal.

Original language | English |
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Pages (from-to) | 583-594 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2015 |

Event | 27th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2015 - Daejeon, Korea, Republic of Duration: Jul 6 2015 → Jul 10 2015 |

### Bibliographical note

Publisher Copyright:© 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

## Keywords

- Algebraic complex
- Discrete Morse Theory
- Homology
- Orthogonality
- Poset decomposition
- Q-analogues

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science
- Discrete Mathematics and Combinatorics