bibkey: mirzavaziri2026gaussianquotient
authors: Madjid Mirzavaziri, Daniel Yaqubi
year: 2026
title: Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
doi: 10.48550/arXiv.2610.19249
url: https://arxiv.org/abs/2610.19249v1
claim: "Problem 7.2 asks whether weak increase of the first Gaussian quotient through its middle forces the two-exception sign pattern after a second q-integer division."
strata_touched:
  - D5/S3/Combinatorics/GaussianQuotient/MirzavaziriYaqubiMonotonicityRefutation
license: citation-only
triage: anchor

Gaussian quotient monotonicity and coefficient signs

Verified locator

DOI: 10.48550/arXiv.2610.19249

URL: https://arxiv.org/abs/2610.19249v1

Problem 7.2 appears on page 16. The source TeX is gk74-2.tex from https://arxiv.org/e-print/2610.19249v1, sha256 8e15a5d14b845dcc082a0a0e819126ca875269aa6f166737d163c7a88a8dd865.

Source statement

The Gaussian bracket below expands the source macro \qbin without changing its typography.

Characterize the quadruples $(N,k,a,b)$ for which ${\left[\genfrac{}{}{0ex}{}{N}{k}\right]}_q/([a{]}_q[b{]}_q)$ is a polynomial whose only negative coefficients are $-1$ at $q^1$ and at $q^{D-1}$. In particular, is this always the case when ${\left[\genfrac{}{}{0ex}{}{N}{k}\right]}_q/[a{]}_q$ is weakly increasing from $q^1$ to its middle coefficient?

The next sentence on page 16 states:

By Lemma 4.1, the answer to the last question is affirmative as soon as, in addition, the coefficients of $q^b$ and $q^{b+1}$ in ${\left[\genfrac{}{}{0ex}{}{N}{k}\right]}_q/[a{]}_q$ are distinct.

The final question is encoded using integer polynomials U and P, positive a,b, k ≤ N, and exact multiplication equalities for the quotients. Weak increase starts at the comparison of coefficients one and two and ends at floor(natDegree U / 2). The exceptional indices are 1 and natDegree P − 1. Natural-index subtraction is truncated at zero. The Gaussian and q-integer definitions are reused from their existing polynomial owners.

Scope of the negative answer

The finite counterexample has (N,k,a,b)=(30,3,7,10), degrees 75 and 66 for U and P, and coefficient −1 at degree 11 in P. It answers the final sufficiency question. The full classification and an unbounded counterexample family are separate questions. Lemma 4.1 retains its additional strictness hypothesis; the counterexample has equal coefficients u10=u11=3.

Exact computations are available in the-omega-institute/trureturing-experiments, commit c9ab263c1e90358645b266282e0d0c168e7f2523, under docs/reports/mirzavaziri-yaqubi-2026-gaussian-quotient-monotonicity-problem-7-2/. The command python3 docs/reports/mirzavaziri-yaqubi-2026-gaussian-quotient-monotonicity-problem-7-2/check.py exited 0 with ALL_CHECKS_OK; check.py has sha256 f67ed55ee555037daf83cc71c4e7a3dc4faea2839d4e00abc80689cfb5528a99. The finite computations do not prove the universal family assertion.

Literature boundary

Issue #21322 records the bounded source and literature screen. No exact settlement was identified there. Completeness of the screen is ASSUMED-UNVERIFIED; Semantic Scholar returned HTTP 429, and several web responses were ineffective.