Proofs of three conjectured internal congruences modulo 32 for Schur-type overpartitions

Abstract

Let S(n) denote the number of overpartitions of Schur-type. Recently, Chern, da Silva and Sellers proved many congruences modulo 8 and 16 for S(n). At the end of their paper, they also posed a conjecture on internal congruences modulo 32 for S(n). In this paper, we confirm their conjecture by using theta function identities and the (p, k)-parameterization of theta functions given by Alaca and Williams. In particular, we show that for any integer j with 0 ≤ j ≤ 31, there are infinitely many integers u_j such that S(u_j ) ≡ j (mod 32).