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 c2,0 and c2,1. They also emphasized that the considerations for the general case of ck,β are important for future work. In this paper, for each k we construct a vector-valued modular form for the generating functions of ck,β, and determine an equivalence relation among all β. Within each equivalence class, we can identify modular transformations relating the congruences of one ck,β to that of another ck,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 cφ3, the Andrews' 3-colored Frobenius partition.
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