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Zeilberger's KOH theorem and the strict unimodality of $ q$-binomial coefficients


Author: Fabrizio Zanello
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
Published electronically: February 6, 2015
MathSciNet review: 3336605
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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.


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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
Email: zanello@math.mit.edu, zanello@mtu.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12510-5
Keywords: $q$-binomial coefficient, Gaussian polynomial, unimodality
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
Article copyright: © Copyright 2015 American Mathematical Society