$q$-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter's solution of the hard-hexagon model

For each $s\in \{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s(\mathrm{m}\mathrm{o}\mathrm{d}10)$, arises naturally in regime III of Rodney Baxter’s solution of the hard-hexagon model of statistical mechanics. For each $s\in \{1,3\}$, the generating function of $R_s^{\ast }(n)$, the number of partitions of $n$ into parts not congruent to $0$, $\pm s(\mathrm{m}\mathrm{o}\mathrm{d}10)$ and $10-2s(\mathrm{m}\mathrm{o}\mathrm{d}20)$, arises naturally in regime IV of Rodney Baxter’s solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $R_s(n)$ and $R_s^{\ast }(n)$, providing new parity results involving sums of partition numbers $p(n)$ and squares in arithmetic progressions.