Arithmetic Properties of Certain t-Regular Partitions

For a positive integer \(t\ge 2\), let \(b_{t}(n)\) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for \(b_9(n)\) and \(b_{19}(n)\). We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of \(b_9(n)\) and \(b_{19}(n)\) modulo 2. For \(t\in \{6,10,14,15,18,20,22,26,27,28\}\), Keith and Zanello conjectured that there are no integers \(A>0\) and \(B\ge 0\) for which \(b_t(An+ B)\equiv 0\pmod 2\) for all \(n\ge 0\). We prove that, for any \(t\ge 2\) and prime \(\ell \), there are infinitely many arithmetic progressions \(An+B\) for which \(\sum _{n=0}^{\infty }b_t(An+B)q^n\not \equiv 0 \pmod {\ell }\). Next, we obtain quantitative estimates for the distributions of \(b_{6}(n), b_{10}(n)\) and \(b_{14}(n)\) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.