Congruence relations for r-colored partitions

Abstract

Let $\ell \ge 5$ be prime. For the partition function $p(n)$ and $5\le \ell \le 31$, Atkin found a number of examples of primes $Q\ge 5$ such that there exist congruences of the form $p(\ell Q^{3}n+\beta )\equiv 0(\mathrm{mod}\ell ).$ Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$. In this paper, for a wide range of $c\in {\mathbb{F}}_{\ell }$, we prove congruences of the form $p(\ell Q^{3}n+{\beta }_0)\equiv c\cdot p(\ell Qn+{\beta }_1)(\mathrm{mod}\ell )$ for infinitely many primes $Q$. For a positive integer $r$, let $p_r(n)$ be the $r$-colored partition function. Our methods yield similar congruences for $p_r(n)$. In particular, if $r$ is an odd positive integer for which $\ell >5r+19$ and $2^{r+2}\not\equiv 2^{\pm 1}(\mathrm{mod}\ell )$, then we show that there are infinitely many congruences of the form $p_r(\ell Q^{3}n+\beta )\equiv 0(\mathrm{mod}\ell )$. Our methods involve the theory of modular Galois representations.