Congruence families modulo powers of $7$ for $4$-colored generalized Frobenius partitions
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908562725273600 |
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| author | Wang, Kangyu Wang, Yining |
| author_facet | Wang, Kangyu Wang, Yining |
| contents | In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for $2$-colored Frobenius partitions $cϕ_2$ 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 $cϕ_3=cψ_{3,3/2}$. In the present paper, we give a new example of Chen and Zhu's framework for $cψ_{4,β}$. Our proof is considerably simpler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruence families modulo powers of $7$ for $4$-colored generalized Frobenius partitions Wang, Kangyu Wang, Yining Combinatorics Primary 11P83, Secondary 05A17, 11F33 In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for $2$-colored Frobenius partitions $cϕ_2$ 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 $cϕ_3=cψ_{3,3/2}$. In the present paper, we give a new example of Chen and Zhu's framework for $cψ_{4,β}$. Our proof is considerably simpler. |
| title | Congruence families modulo powers of $7$ for $4$-colored generalized Frobenius partitions |
| topic | Combinatorics Primary 11P83, Secondary 05A17, 11F33 |
| url | https://arxiv.org/abs/2509.23237 |