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

TL;DR

This paper explores $q$-series congruences related to partition functions from Baxter's solution of the hard-hexagon model, revealing new parity properties and connections to partition numbers and squares in arithmetic progressions.

Contribution

It introduces novel parity results for partition functions arising in statistical mechanics, linking them to classical partition numbers and quadratic sums.

Findings

New parity results for $R_s(n)$ and $R^*_s(n)$

Connections between partition functions and sums of squares

Insights into $q$-series congruences in statistical mechanics

Abstract

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 $0$, $\pm s\pmod {10}$, 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 $R^*_s(n)$, the number of partitions of $n$ into parts not congruent to $0$, $\pm s\pmod {10}$ and $10-2s \pmod {20}$, 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 $R^*_s(n)$, providing new parity results involving sums of partition numbers $p(n)$ and squares in arithmetic progressions.