Analytic q-inequalities and a proof of Bergeron's conjecture
For integers $1\le a\le b\le c\le d$ with $ad=bc$, Bergeron asked whether the difference of Gaussian binomial coefficients $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ has nonnegative coefficients. We prove that it does, and more generally that $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ has nonnegative coefficients whenever $a\le b\le c$ and $ad\le bc$, a strengthening proposed by Troyka. The proof is analytic rather than bijective. We develop a toolkit for coefficientwise inequalities, including a coefficientwise local limit theorem for Gaussian polynomials based on Irwin--Hall densities, and use it in a…
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