The difference of the sums of odd and even parts of restricted partitions

Abstract

For a subset $D \subset \mathbb{N}$, denote by $S\left(D,n\right)$ the total sum of all odd parts minus the sum of all even parts of all partitions of $n$ in which parts from $D$ do not repeat. In this paper, we derive the generating function for $S\left(2\mathbb{N},n\right)$ and use it to give some congruences modulo 4. We also derive the generating function for $S\left(2\mathbb{N}-1,n\right)$ and give two congruences for these functions, one of which was previously proved by Garvan and Sarma.