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 alpha(24z) of a certain weak Maass form M(z) of weight 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(p^2) for primes p > 3, and an eigenform modulo t for t = 5, 7 or 13 provided that (t, 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 t (namely 5^6, 7^4 and 13^2) occur for the coefficients of the function alpha(z). Analogous results for the partition function were found by Atkin in 1966. Our results depend on the recent result of Ono that M[p](z/24) is a weakly holomorphic modular form of weight 3/2 for the full modular group where M[p](z) = M(z)|T(p^2) - chi(p)(1 + p)M(z).