Congruence properties modulo prime powers for a class of partition functions

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1 - q^{pn})^{-1}$. This function and its generalizations $p_{[c^{\ell}, d^m]}(n)$ are the subject of study in several recent papers. Let $\ell\geq 5$, let $j\geq 1$, and let $p \in \{2, 3, 5\}$. In this paper, we prove that the generating function for $p_{[1, p]}(n)$ in the progression $\beta_{p, \ell, j}$ modulo $\ell^j$ with $24\beta_{p, \ell, j} \equiv p + 1 \pmod{\ell^j}$ lies in a Hecke-invariant subspace of type $\{\eta(Dz)\eta(Dpz)F(Dz) : F(z) \in M_{s}(\Gamma_0(p), \chi)\}$ for suitable $D\geq 1$, $s\geq 0$, and character~$\chi$. When $p\in \{2, 3, 5\}$, we use the Hecke-invariance of these subspaces proved in [21] to prove, for distinct primes $\ell$ and $m\geq 5$ and $j\geq 1$, congruences of the form $$p_{[1, p]}\left(\frac{\ell^jm^k n + 1}{D}\right)\equiv 0 \pmod{\ell^j}$$ for all $n\geq 1$ with $m\nmid n$, where $k$ is explicitly computable and depends on the forms in the invariant subspace. Our proofs require adapting and extending analogous level one results on $p(n)$ in [1] and [22] to level $p$.