Arithmetic properties of generalized Frobenius partitions

Abstract

Ramanujan proved three famous congruences for the partition function modulo 5, 7, and 11. The first author and Boylan proved that these congruences are the only ones of this type. In 1984 Andrews introduced the $m$-colored Frobenius partition functions $c\phi_m$; these are natural higher-level analogues of the partition function which have attracted a great deal of attention in the ensuing decades. For each $m\in \{5, 7, 11\}$ there are two analogues of Ramanujan's congruences for $c\phi_m$, and for these $m$ we prove there are no congruences like Ramanujan's other than these six. Our methods involve a blend of theory and computation with modular forms.