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Congruences modulo powers of $7$ for partitions with colored odd parts

  • Saikat Maity
  • Manjil P. Saikia
  • Prabal Talukdar
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Abstract

Let ak(n) denote the number of partitions of n in which even parts occur in one color and odd parts may occur in k colors. We prove infinite families of congruences modulo powers of $7$ for ak(n) when k ∈ {4,5,7}. For each positive integer α, let λk,α be the least nonnegative solution of 24λk,α≡−(k−2)pmod7α. Then, we show that ak(7αn+λk,α) ≡ 0pmod7⌊(α+1)/2⌋ for every nonnegative integer n. These results extend congruences modulo $7$ of Hirschhorn and Sellers (2026) and complement Chen's (2025) congruences for the reciprocal crank parity function a3(n).

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Cite

BibTeXRIS
BibTeX
@misc{maity2026congruences,
  title = {{Congruences modulo powers of $7$ for partitions with colored odd parts}},
  author = {Saikat Maity and Manjil P. Saikia and Prabal Talukdar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35292},
  eprint = {2610.35292},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35292},
}
RIS
TY  - UNPB
TI  - Congruences modulo powers of $7$ for partitions with colored odd parts
AU  - Saikat Maity
AU  - Manjil P. Saikia
AU  - Prabal Talukdar
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35292
UR  - https://arxiv.org/abs/2610.35292
AB  - Let a_k(n) denote the number of partitions of n in which even parts occur in one color and odd parts may occur in k colors. We prove infinite families of congruences modulo powers of $7$ for a_k(n) when k ∈ {4,5,7}. For each positive integer α, let λ_k,α be the least nonnegative solution of 24λ_k,α≡−(k−2)pmod7^α. Then, we show that a_k(7^αn+λ_k,α) ≡ 0pmod7^⌊(α+1)/2⌋ for every nonnegative integer n. These results extend congruences modulo $7$ of Hirschhorn and Sellers (2026) and complement Chen's (2025) congruences for the reciprocal crank parity function a_3(n).
ER  -