Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
Abstract
We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer n, the polynomial Pn(q) = ((1−q)2)/((1−q10n−1)(1−q15n−1))qbin30n5n has non-negative coefficients, except for q1 and q125n2−25n+3, which equal −1. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last 25n+1 coefficients via a novel partition injection. Second, by introducing a ``defect'' generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large n ≥ 2· 1029. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.
Related
- The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index
- The Sufficient Set of Congruences of Binomial Coefficients for Primality
- Infinitely Many Binomial Coefficients of Deficiency One
- Binomial coefficients and the operadic butterfly
- Multiple harmonic star sums involving central binomial coefficients as weights
BibTeX
@misc{mirzavaziri2026graph,
title = {{Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler}},
author = {Madjid Mirzavaziri and Daniel Yaqubi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19249},
eprint = {2610.19249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19249},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
AU - Madjid Mirzavaziri
AU - Daniel Yaqubi
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19249
UR - https://arxiv.org/abs/2610.19249
AB - We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer n, the polynomial P_n(q) = ((1−q)^2)/((1−q^10n−1)(1−q^15n−1))qbin30n5n has non-negative coefficients, except for q^1 and q^125n^2−25n+3, which equal −1. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last 25n+1 coefficients via a novel partition injection. Second, by introducing a ``defect'' generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large n ≥ 2· 10^29. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.
ER -