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Coefficientwise positivity for alternating sums of qq-factorials

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by David Anderson ↗

SubmitterDavid Anderson

Version 1 / Sep 30, 2026 / CC0 1.0

Abstract

We prove that the alternating sum ∑i=0k(−1)i(ki)[m−i]!q\sum_{i=0}^{k}(-1)^i\binom ki[m-i]!_q has nonnegative coefficients whenever m≥2k−1m\ge 2k-1. For k≥3k\ge3, this range is exact. The case m=2km=2k proves a conjecture of Lewis and Morales arising from the enumeration of invertible matrices with prescribed zero entries. After removing a common qq-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.

Provenance statement

This work was done entirely by LLMs. At least one human (the contributer) has read and understood the proof (which is not complicated). Further details, including transcripts of prompts and reponses, are posted here: https://github.com/pseudoeffective/opac012 . The same GitHub repo hosts a Lean formalization. Further context, including motivation for undertaking this particular project is in this essay: https://pseudoeffective.github.io/essays/mathematics_in_the_dark_forest.html .

Tools used

Anthropic
Claude FableVersion 5.1
OpenAI
CodexVersion 6
Lean
LeanVersion 4

References

  1. J. B. Lewis and A. H. Morales, Rook theory of the finite general linear group , Experiment. Math. 29 (2020), no. 3, 328--346; https://arxiv.org/abs/1707.08192 arXiv:1707.08192 .arXiv
  2. J. B. Lewis, Open Problems in Algebraic Combinatorics , blog maintained by S. Hopkins, https://realopacblog.wordpress.com/2019/09/30/matrix-counting-over-finite-fields/ realopacblog.wordpress.com/2019/09/30/matrix-counting-over-finite-fields/ (2019).link

Version history

  1. v1Initial depositCurrentSep 30, 2026