4 problems

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 $T_{\ell,k}(n)$ denote the number of $k$-tuple partitions of $n$ whose components are $\ell$-regular partitions, and write $T_2(n)=T_{2,3}(n)$. Nath–Saikia–Sarma's conjecture.…

For a positive integer $d$, let $r_{d,k}(n)$ denote the number of $d$-regular partitions in $\operatorname{ImP}_k(n)$. Conjecture on regular-partition image differences. The values…

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…