math.CO — Combinatorics

Coefficientwise positivity for alternating sums of $q$-factorials

Contributed by David Anderson

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.