Proof of a nonnegativity conjecture of Prellberg--Stanton
Abstract
In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series q+(1−q)∏i=n2n−11/(1−qi) are nonnegative for n≥3 and n being odd. Let qbinom nm denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series (1−q)qbinom nm∏i=0m−11/(1−qi+m) are nonnegative for 0<m<n, except when n is even and m∈{2,n−2}. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.
Related
BibTeX
@misc{ai2026proof,
title = {{Proof of a nonnegativity conjecture of Prellberg--Stanton}},
author = {Hong-Zhang Ai and Lili Mu and Bao-Xuan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18377},
eprint = {2610.18377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18377},
note = {CC0},
}
RIS
TY - UNPB
TI - Proof of a nonnegativity conjecture of Prellberg--Stanton
AU - Hong-Zhang Ai
AU - Lili Mu
AU - Bao-Xuan Zhu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18377
UR - https://arxiv.org/abs/2610.18377
AB - In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series q+(1−q)∏_i=n^2n−11/(1−q^i) are nonnegative for n≥3 and n being odd. Let qbinom nm denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series (1−q)qbinom nm∏_i=0^m−11/(1−q^i+m) are nonnegative for 0<m<n, except when n is even and m∈{2,n−2}. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.
ER -