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 is an element of {2, 4}, the generating function of R-s (n), the number of partitions of n into odd parts or congruent to 0, +/- s (mod 10), arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. For each s is an element of {1, 3}, the generating function of R-s*(n), the number of partitions of n into parts not congruent to 0, +/- s (mod 10) and 10 - 2s (mod 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.