Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences

TL;DR

This paper discovers new congruences for Andrews' spt-function using weak Maass forms and Hecke operators, extending Atkin's classical partition congruences to higher powers and different moduli.

Contribution

It establishes that the generating function for spt(n) is an eigenform modulo various primes and powers, extending known congruences and linking to recent modular form results.

Findings

spt(n) generating function is an eigenform modulo 72 for Hecke operators

New congruences for spt(n) modulo 5, 7, 13 and their powers

Analogous results extend Atkin's classical 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).