The Andrews-Stanley partition function and $p(n)$: congruences

Abstract:

R. Stanley formulated a partition function $t(n)$ which counts the number of partitions $\pi$ for which the number of odd parts of $\pi$ is congruent to the number of odd parts in the conjugate partition $\pi ’$ (mod 4). In light of G. E. Andrews’ work on this subject, it is natural to ask for relationships between $t(n)$ and the usual partition function $p(n)$. In particular, Andrews showed that the (pmod 5) Ramanujan congruence for $p(n)$ also holds for $t(n)$. In this paper we extend his observation by showing that there are infinitely many arithmetic progressions $An + B$ such that for all $n\geq 0$, \[ t(An+B) \equiv p(An+B) \equiv 0 \pmod {l^j} \] whenever $l\geq 5$ is prime and $j\geq 1$.