Congruences for powers of the partition function

TL;DR

This paper explores congruences for powers of the partition function, extending known results and establishing new infinite families of such congruences, while also identifying cases where they do not hold.

Contribution

It generalizes and extends previous work on partition function congruences, providing new infinite families and non-existence results for certain primes and parameters.

Findings

Established new infinite families of congruences for $p_{-t}(n)$.

Identified conditions under which such congruences do not exist.

Extended Ramanujan-type congruence results to colored partition functions.

Abstract

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such congruences, those of the form $p_{-t}(\ell n + a) \equiv 0 \pmod {\ell}$, were previously studied by Kiming and Olsson. If $\ell \geq 5$ is prime and $-t \not \in \{\ell - 1, \ell -3\}$, then such congruences satisfy $24a \equiv -t \pmod {\ell}$. Inspired by Lin's example, we obtain natural infinite families of such congruences. If $\ell\equiv2\pmod{3}$ (resp. $\ell\equiv3\pmod{4}$ and $\ell\equiv11\pmod{12}$) is prime and $r\in\{4,8,14\}$ (resp. $r\in\{6,10\}$ and $r=26$), then for $t=\ell s-r$, where $s\geq0$, we have that \begin{equation*} p_{-t}\left(\ell n+\frac{r(\ell^2-1)}{24}-\ell\Big\lfloor\frac{r(\ell^2-1)}{24\ell}\Big\rfloor\right)\equiv0\pmod{\ell}. \end{equation*} Moreover, we exhibit infinite families where such congruences cannot hold.