Congruence relations for r-colored partitions

Abstract

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$ 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 $p(\ell Q^{3} n+\beta_{0}) \equiv c \cdot p(\ell Q n+\beta_{1}) \pmod{\ell}$ 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 $2^{r+2} \not \equiv 2^{\pm 1} \pmod{\ell}$, then we show that there are infinitely many congruences of the form $p_{r}(\ell Q^{3}n+\beta) \equiv 0 \pmod{\ell}$. Our methods involve the theory of modular Galois representations.