3 problems

Let $b_{3,e}(n)$ and $b_{3,o}(n)$ denote the numbers of $3$-regular partitions of $n$ having, respectively, an even and an odd number of parts. Let $n\geqslant 0$, let…

Let $T_{\ell,k}(n)$ denote the number of $k$-tuple partitions of $n$ whose components are $\ell$-regular partitions, and write $T_2(n)=T_{2,3}(n)$. Let $p\geq 5$ be a prime satisfy…

Let $b_k(n)$ denote the number of $k$-regular partitions of $n$, where $k\in\{3,5\}$, and let $m$ be a positive integer. A Ramanujan-type congruence is a congruence of the form … f…