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A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and t-Core Partitions
Abstract
Let at(n) denote the coefficients of the Nekrasov--Okounkov hook length generating function Ft2(x), and let bt(n) denote the coefficients of the t-core partition generating function Ct(x). We prove a sharp prime-power dichotomy: for every integer t≥ 2, the congruence at(n)≡ bt(n)pmod t holds for all n≥ 0 if and only if t is a prime power. When t=p is prime, the congruence strengthens to modulo p2. For t=6, and more generally for every integer that is not a prime power, the congruence fails. This identifies prime powers as the exact moduli for which these two partition-theoretic coefficient sequences are arithmetically equivalent.
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BibTeX
@misc{sharma2026primepower,
title = {{A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and $t$-Core Partitions}},
author = {Meenu Sharma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01422},
eprint = {2610.01422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01422},
}
RIS
TY - UNPB
TI - A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and t-Core Partitions
AU - Meenu Sharma
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01422
UR - https://arxiv.org/abs/2610.01422
AB - Let a_t(n) denote the coefficients of the Nekrasov--Okounkov hook length generating function F_t^2(x), and let b_t(n) denote the coefficients of the t-core partition generating function C_t(x). We prove a sharp prime-power dichotomy: for every integer t≥ 2, the congruence a_t(n)≡ b_t(n)pmod t holds for all n≥ 0 if and only if t is a prime power. When t=p is prime, the congruence strengthens to modulo p^2. For t=6, and more generally for every integer that is not a prime power, the congruence fails. This identifies prime powers as the exact moduli for which these two partition-theoretic coefficient sequences are arithmetically equivalent.
ER -