On Bergeron's positivity problem for \(q\)-binomial coefficients
The electronic journal of combinatorics, Tome 25 (2018) no. 2
F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$ has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.
DOI :
10.37236/7358
Classification :
05A15, 05A17, 05A20
Mots-clés : \(q\)-binomial coefficient, unimodality, positivity, KOH theorem
Mots-clés : \(q\)-binomial coefficient, unimodality, positivity, KOH theorem
Affiliations des auteurs :
Fabrizio Zanello 1
@article{10_37236_7358,
author = {Fabrizio Zanello},
title = {On {Bergeron's} positivity problem for \(q\)-binomial coefficients},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/7358},
zbl = {1391.05035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7358/}
}
Fabrizio Zanello. On Bergeron's positivity problem for \(q\)-binomial coefficients. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/7358
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