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 $c{\psi }_{2,0}$ and $c{\psi }_{2,1}$. They also emphasized that the considerations for the general case of $c{\psi }_{k,\beta }$ are important for future work. In this paper, for each $k$ we construct a vector-valued modular form for the generating functions of $c{\psi }_{k,\beta }$, and determine an equivalence relation among all $\beta$. Within each equivalence class, we can identify modular transformations relating the congruences of one $c{\psi }_{k,\beta }$ to that of another $c{\psi }_{k,{\beta }^{\prime }}$. 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 $c{\varphi }_3$, the Andrews' $3$-colored Frobenius partition.

54 pages