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