Ramanujan-type congruences for the smallest parts function

Abstract

In a famous paper of Andrews, he defined and studied the smallest parts function $\operatorname{spt}(n)$. He showed that it satisfies the three Ramanujan-type congruences: \[ \operatorname{spt}(5n+4)\equiv0\pmod5,\qquad \operatorname{spt}(7n+5)\equiv0\pmod7,\qquad \operatorname{spt}(13n+6)\equiv0\pmod{13}. \] In a recent preprint, the author showed that these are the only "canonical'' congruences. Here, we complete the classification of Ramanujan-type congruences for $\operatorname{spt}$, showing that if $\operatorname{spt}(\ell n+b)\equiv0\pmod\ell$ for all $n\geq0$, where $\ell\geq5$ is prime, then $(\ell,b)\in\{(5,4),(7,5),(13,6)\}.$ That is, Andrews' congruences are the only Ramanujan-type congruences for $\operatorname{spt}(n)$. These results have been verified in Lean by AxiomProver.