3 problems

Let $p(n)$ denote the partition function and let ${\mathrm{spt}}(n)$ denote the smallest-parts function, which counts the smallest parts among the partitions of $n$. Chen's conject…

Let $\operatorname{spt}(n)$ denote the number of occurrences of the smallest part among the partitions of $n$. Let $\ell>3$ be prime with $\ell\not\equiv23\pmod{24}$, and define ……

For each $n$, let $N_S(m,n)$ denote the number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Dyson and Rhoades's conjecture. The sequence $\{N_S(m,n)\}_{m}$ is weakly unimo…