New congruences involving statistical mechanics partition functions in regime III and IV of Baxter’s solution of the hard-hexagon model

Abstract

Recently, Merca presented characterizations of the parities on $R_s(n)$ and $R_t^{*}(n)$ with $s\in \{2,4\}$ and $t\in \{1,3\}$, where $R_s(n)$ counts the number of partitions of n into odd parts or congruent to $0, \pm s\ (\textrm{mod}\ 10)$ and $R_t^{*}(n)$ counts the number of partitions of n into parts not congruent to $0, \pm t\ (\textrm{mod} 10)$ and $10 \pm 2t\ (\textrm{mod} \ 20)$. The generating function of $R_s(n)$ arises naturally in regime III of Baxter’s solution of the hard-hexagon model of statistical mechanics and the generating function of $R_t^{*} (n)$ arises naturally in regime IV of Baxter’s solution of the hard-hexagon model of statistical mechanics. In this paper, we prove some congruences modulo 4 and 8. In particular, we establish several infinite families of congruences modulo 4 and 8 for $R_s(n)$ and $R_t^{*}(n)$ and prove that $\frac{R_s(n)}{8}$ and $\frac{R_t^{*}(n)}{8}$ take integer values with probability 1 for $n\ge 0$.