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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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.
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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:,
  • 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:
  • MathSciNet review: 3336605