Correspondence among congruence families for generalized Frobenius partitions via modular permutations

2025/06/20 by Chen, Rong, Zhu, Xiao-Jie · 2 citations
Mathematics · #11F33 #11F37 #20C15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11P83 #Representation Theory (math.RT) #Secondary 11F27

paper · pdf · doi:10.48550/arxiv.2506.16823

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 cϕ₃, the Andrews' 3-colored Frobenius partition.

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• Congruences modulo powers of 3 for generalized Frobenius partitions CΨ_6,0
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