Congruences for powers of the partition function

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