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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Zeilberger’s KOH theorem and the strict unimodality of $q$-binomial coefficients
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by Fabrizio Zanello PDF
Proc. Amer. Math. Soc. 143 (2015), 2795-2799 Request permission

Abstract:

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\binom {a+b}{b}_q$. Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger’s KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\binom {a+b}{b}_q$ can get arbitrarily large, when we assume that $b$ is fixed and $a$ is large enough.
References
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Additional Information
  • Fabrizio Zanello
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 — and — Department of Mathematical Sciences, Michigan Tech, Houghton, Michigan 49931-1295
  • MR Author ID: 721303
  • Email: zanello@math.mit.edu, zanello@mtu.edu
  • Received by editor(s): November 18, 2013
  • Received by editor(s) in revised form: February 10, 2014
  • Published electronically: February 6, 2015
  • Communicated by: Jim Haglund
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2795-2799
  • MSC (2010): Primary 05A15; Secondary 05A17
  • DOI: https://doi.org/10.1090/S0002-9939-2015-12510-5
  • MathSciNet review: 3336605