Congruence families modulo powers of 7 for 4-colored generalized Frobenius partitions

2025/09/27 by Wang, Kangyu, Wang, Yining · 1 citation
#11F33 #Combinatorics (math.CO) #FOS: Mathematics #Primary 11P83 #Secondary 05A17

paper · doi:10.48550/arxiv.2509.23237

Abstract

In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions cϕ₂ introduced by George Andrews. Recently, Frank Garvan, James Sellers and Nicolas Smoot showed that this family of congruences is equivalent to the family of congruences for (2,0)-colored Frobenius partitions cψ_2,0 introduced by Brian Drake and by Yuze Jiang, Larry Rolen and Michael Woodbury for the general case. Motivated by Garvan, Sellers and Smoot's work, Rong Chen and Xiao-Jie Zhu found modular transformations relating the cψ_k,β for fixed k and varying β. As an example, they proved a family of congruences for cψ_3,1/2 following Paule and Radu's work and then proved the equivalence between cψ_3,1/2 and cϕ₃=cψ_3,3/2. In the present paper, we give a new example of Chen and Zhu's framework for cψ_4,β. Our proof is considerably simpler.

Citations

• Correspondence among congruence families for generalized Frobenius partitions via modular permutations
• Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta functions
• Generalized Frobenius partitions, Motzkin paths, and Jacobi forms
• A tutorial for the MAPLE ETA package

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

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