Arithmetic properties of certain $t$-regular partitions

Abstract

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 $2$. We also relate $b_t(n)$ to the ordinary partition function, and prove that $b_t(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. 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 $b_t(An+B)\equiv 0(\mathrm{mod}2)$ 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 ${\sum }_{n=0}^{\infty }{b}_t(An+B)q^n\not\equiv 0(\mathrm{mod}\ell )$. 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.

17 pages