Congruences for powers of the partition function

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 $p_{-3}(11n+7)\equiv 0(\mathrm{mod}11)$ for every integer $n$. Such congruences, those of the form $p_{-t}(\ell n+a)\equiv 0(\mathrm{mod}\ell )$, were previously studied by Kiming and Olsson. If $\ell \ge 5$ is prime and $-t\notin \{\ell -1,\ell -3\}$, then such congruences satisfy $24a\equiv -t(\mathrm{mod}\ell )$. Inspired by Lin's example, we obtain natural infinite families of such congruences. If $\ell \equiv 2(\mathrm{mod}3)$ (resp. $\ell \equiv 3(\mathrm{mod}4)$ and $\ell \equiv 11(\mathrm{mod}12)$) is prime and $r\in \{4,8,14\}$ (resp. $r\in \{6,10\}$ and $r=26$), then for $t=\ell s-r$, where $s\ge 0$, 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.