Congruences modulo powers of $7$ for partitions with colored odd parts

Abstract

Let $a_k(n)$ denote the number of partitions of $n$ in which even parts occur in one color and odd parts may occur in $k$ colors. We prove infinite families of congruences modulo powers of $7$ for $a_k(n)$ when $k\in \{4,5,7\}$. For each positive integer $\alpha$, let ${\lambda }_{k,\alpha }$ be the least nonnegative solution of $24{\lambda }_{k,\alpha }\equiv -(k-2)(\mathrm{mod}{7}^{\alpha })$. Then, we show that \[ a_k(7^αn+λ_{k,α}) \equiv 0\pmod{7^{\lfloor(α+1)/2\rfloor}} \] for every nonnegative integer $n$. These results extend congruences modulo $7$ of Hirschhorn and Sellers (2026) and complement Chen's (2025) congruences for the reciprocal crank parity function $a_3(n)$.

arXiv comment24 pages