Bergeron's conjecture & a tale of two binomial coefficients

Abstract

Bergeron's conjecture states that, if $1\le a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\left(\genfrac{}{}{0ex}{}{b+c}{b}\right)}_q\ge {\left(\genfrac{}{}{0ex}{}{a+d}{a}\right)}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.

8 pages