PreprintCC BY 4.0arXiv
A positivity conjecture of Prellberg and Stanton
Abstract
We prove the conjecture of Prellberg and Stanton (2003) that (1−q)/(qn;q)mgenfrac[]0ptnmq has nonnegative power-series coefficients for 0<m<n, except when n is even and m∈{2,n−2}. We obtain a polynomial version by bounding a Möbius correction by first differences of Gaussian coefficients. The proof uses partitions and quadratic and cubic semi-invariants of binary forms. An extension of this bound proves the Reiner--Stanton--White conjecture on positivity of orbit polynomials for flags of subspaces.
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BibTeX
@misc{prellberg2026positivity,
title = {{A positivity conjecture of Prellberg and Stanton}},
author = {Thomas Prellberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26878},
eprint = {2610.26878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A positivity conjecture of Prellberg and Stanton
AU - Thomas Prellberg
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.26878
UR - https://arxiv.org/abs/2610.26878
AB - We prove the conjecture of Prellberg and Stanton (2003) that (1−q)/(q^n;q)_mgenfrac[]0ptnm_q has nonnegative power-series coefficients for 0<m<n, except when n is even and m∈{2,n−2}. We obtain a polynomial version by bounding a Möbius correction by first differences of Gaussian coefficients. The proof uses partitions and quadratic and cubic semi-invariants of binary forms. An extension of this bound proves the Reiner--Stanton--White conjecture on positivity of orbit polynomials for flags of subspaces.
ER -