Negative q-Stirling numbers

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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 languageEnglish
Pages (from-to)583-594
Number of pages12
JournalDiscrete Mathematics and Theoretical Computer Science
StatePublished - 2015
Event27th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2015 - Daejeon, Korea, Republic of
Duration: Jul 6 2015Jul 10 2015

Bibliographical note

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


  • 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


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