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

$${p_{-t}}(\ell n + \frac{r(\ell^{2}-1)}{24}-\ell\lfloor\frac{r(\ell^{2}-1)}{24\ell}\rfloor)\equiv0 \,\,({\rm mod} \ell).$$

Moreover, we exhibit infinite families where such congruences cannot hold.