q-Stirling numbers: A new view

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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 languageEnglish
Pages (from-to)50-80
Number of pages31
JournalAdvances in Applied Mathematics
Volume86
DOIs
StatePublished - 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

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