Correspondence among congruence families for generalized Frobenius partitions via modular permutations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910014952701952 |
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| author | Chen, Rong Zhu, Xiao-Jie |
| author_facet | Chen, Rong Zhu, Xiao-Jie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Correspondence among congruence families for generalized Frobenius partitions via modular permutations Chen, Rong Zhu, Xiao-Jie Number Theory Combinatorics Representation Theory Primary 11P83, Secondary 11F27, 11F33, 11F37, 20C15 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. |
| title | Correspondence among congruence families for generalized Frobenius partitions via modular permutations |
| topic | Number Theory Combinatorics Representation Theory Primary 11P83, Secondary 11F27, 11F33, 11F37, 20C15 |
| url | https://arxiv.org/abs/2506.16823 |