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PreprintCC BY 4.0arXiv

A uniform proof approach for congruences modulo 3 for partitions with k-colored odd parts

  • Marie Jameson
  • James A. Sellers
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Abstract

In recent work, Hirschhorn and the second author defined ak(n) to be the number of partitions of n wherein the even parts come in only one color, while the odd parts may be ``colored'' with one of k colors for fixed k≥ 1. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying ak(n) has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.

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Cite

BibTeXRIS
BibTeX
@misc{jameson2026uniform,
  title = {{A uniform proof approach for congruences modulo 3 for partitions with $k$-colored odd parts}},
  author = {Marie Jameson and James A. Sellers},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19440},
  eprint = {2609.19440},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19440},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A uniform proof approach for congruences modulo 3 for partitions with k-colored odd parts
AU  - Marie Jameson
AU  - James A. Sellers
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.19440
UR  - https://arxiv.org/abs/2609.19440
AB  - In recent work, Hirschhorn and the second author defined a_k(n) to be the number of partitions of n wherein the even parts come in only one color, while the odd parts may be ``colored'' with one of k colors for fixed k≥ 1. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying a_k(n) has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.
ER  -