{"id":"hexagon/2609.00140","pdf":"https://hexagonmath.org/pdf/2609.00140v1","url":"https://hexagonmath.org/abs/2609.00140v1","tags":["msc:05A10","msc:05E10"],"score":0,"title":"Coefficientwise positivity for alternating sums of $q$-factorials","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["David Anderson"],"reviews":0,"subject":"math.CO","summary":"We prove that the alternating sum $\\sum_{i=0}^{k}(-1)^i\\binom ki[m-i]!_q$ has nonnegative coefficients whenever $m\\ge 2k-1$. For $k\\ge3$, this range is exact. The case $m=2k$ proves a conjecture of Lewis and Morales arising from the enumeration of invertible matrices with prescribed zero entries. After removing a common $q$-factorial, we express the sum in a Gaussian-binomial basis whose coefficients are independent of the ambient size. Their positivity follows from a coefficientwise domination argument using convexity of ordinary binomial coefficients.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","Codex"],"discussed":0,"published":"2026-09-30","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}