Parity results for 9-regular partitions

Let t≥2 be an integer. We say that a partition is t-regular if none of its parts is divisible by t, and denote the number of t-regular partitions of n by b _t (n). In this paper, we establish several infinite families of congruences modulo 2 for b ₉(n). For example, we find that for all integers n≥0 and k≥0, $$b_9 \biggl(2^{6k+7}n+ \frac{2^{6k+6}-1}{3} \biggr)\equiv 0 \quad (\mathrm{mod}\ 2 ). $$