Correspondence among congruence families for generalized Frobenius partitions via modular permutations

Abstract

In 2024, Garvan, Sellers and Smoot discovered a remarkable symmetry in the families of congruences for generalized Frobenius partitions $cψ_{2,0}$ and $cψ_{2,1}$. They also emphasized that the considerations for the general case of $cψ_{k,β}$ are important for future work. In this paper, for each $k$ we construct a vector-valued modular form for the generating functions of $cψ_{k,β}$, and determine an equivalence relation among all $β$. Within each equivalence class, we can identify modular transformations relating the congruences of one $cψ_{k,β}$ to that of another $cψ_{k,β'}$. Furthermore, correspondences between different equivalence classes can also be obtained through linear combinations of modular transformations. As an example, with the aid of these correspondences, we prove a family of congruences of $cϕ_{3}$, the Andrews' $3$-colored Frobenius partition.