Congruence properties modulo prime powers for a class of partition functions
Matthew Boylan, Swati
TL;DR
The paper extends the modular-form approach used for Ramanujan-type congruences of the partition function to the two-color partition function p_{[1,p]}(n) with level p, establishing that its generating functions mod \\ell^j lie in Hecke-invariant eta-quotient subspaces. It constructs explicit modular forms modulo \\ell^j and leverages Hecke invariance at index \\ell^2 to produce infinite families of congruences p_{[1,p]}(\\ell^j m^k n + 1)/D \\equiv 0 \\pmod{\\ell^j} for distinct primes \\ell,m \\ge 5, with computable exponents k and explicit constants derived from matrix orders. The method parallels level-one results for p(n) but requires adapting level-p machinery, including cusp-order controls and dimension bounds, to obtain concrete congruences (and their explicit J,N) in the p_{[1,p]} context. These results deepen the connection between partition arithmetic and the theory of modular forms by showing that eta-quotient-invariant subspaces can govern congruences modulo prime powers for more general colored-partition generating functions.
Abstract
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 $β_{p, \ell, j}$ modulo $\ell^j$ with $24β_{p, \ell, j} \equiv p + 1 \pmod{\ell^j}$ lies in a Hecke-invariant subspace of type $\{η(Dz)η(Dpz)F(Dz) : F(z) \in M_{s}(Γ_0(p), χ)\}$ for suitable $D\geq 1$, $s\geq 0$, and character~$χ$. 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 $p$.