A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and $t$-Core Partitions

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 $t\ge 2$, the congruence $$a_t(n)\equiv b_t(n)(\mathrm{m}\mathrm{o}\mathrm{d}t)$$ holds for all $n\ge 0$ 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.