Yue Cai ; Margaret Readdy - Negative $q$-Stirling numbers

dmtcs:2503 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2503
Negative $q$-Stirling numbersArticle

Authors: Yue Cai ORCID1; Margaret Readdy ORCID1

  • 1 Department of Mathematics

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.


Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: $q$-analogues,discrete Morse Theory,poset decomposition,algebraic complex,homology,orthogonality,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

Consultation statistics

This page has been seen 243 times.
This article's PDF has been downloaded 322 times.