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 \(\ell \ge 5\) , let \(j\ge 1\) , 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 \(24\beta _{p, \ell , j} \equiv p + 1\ (\textrm{mod}\ \ell ^j)\) 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 \(D\ge 1\) , \(s\ge 0\) , 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 \(m\ge 5\) and \(j\ge 1\) , congruences of the form \(\begin{aligned} p_{[1, p]}\left( \frac{\ell ^jm^k n +1}{D}\right) \equiv 0\ (\textrm{mod}\ \ell ^j) \end{aligned}\) for all \(n\ge 1\) with \(m\not \mid n\) , 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.
