Congruences for Andrews’ spt-Function Modulo 32760 and Extension of Atkin’s Hecke-Type Partition Congruences

Abstract

New congruences are found for Andrews’ smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part α(24z) of a certain weak Maass form $\mathcal{M}(z)$ of weight $\tfrac{3} {2}$. We show that a normalized form of the generating function for spt(n) is an eigenform modulo 72 for the Hecke operators T(ℓ ²) for primes ℓ > 3 and an eigenform modulo p for p = 5, 7, or 13 provided that (ℓ, 6p) = 1. The result for the modulus 3 was observed earlier by the author and considered by Ono and Folsom. Similar congruences for higher powers of p (namely 5⁶, 7⁴, and 13²) occur for the coefficients of the function α(z). Analogous results for the partition function were found by Atkin in 1966. Our results depend on the recent result of Ono that $\mathcal{M}_{\ell}(z/24)$ is a weakly holomorphic modular form of weight $\tfrac{3} {2}$ for the full modular group where

$$\displaystyle{\mathcal{M}_{\ell}(z) = \mathcal{M}(z)\vert T{(\ell}^{2}) -{ 3\overwithdelims() \ell} (1+\ell)\mathcal{M}(z).}$$