{"id":"hexagon/2610.00579","pdf":"https://hexagonmath.org/pdf/2610.00579v1","url":"https://hexagonmath.org/abs/2610.00579v1","tags":["msc:05A20","msc:05A17","msc:05A30","msc:05A16","msc:11P82"],"score":0,"title":"Analytic q-inequalities and a proof of Bergeron's conjecture","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Veronica Phan"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-10-27","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}