A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$-series

Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$ which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors.

They obtained a complete description modulo 4 for $s_1(n)$.

They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$.

At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$.

In this paper, we confirm this conjecture.

Moreover,we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that $s_1(n)/8$ takes integer values with natural density 1 for $n\geq 0$.