Hecke-type congruences for two smallest parts functions

Abstract

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\bar{p}(n)$ and two smallest parts functions: $\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$ for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function $p(n)$ in 1966 and Garvan for the smallest parts function $\operatorname{spt}(n)$ in 2010. The proofs depend on congruences between the generating functions for $\bar{p}(n)$, $\bar{\operatorname{spt1}}(n),$ and $\operatorname{M2spt}(n)$ and eigenforms for the half-integral weight Hecke operator $T(\ell^{2})$.