Efficient Computation and Congruences for Colored Partition Functions
Abstract
For a positive integer α, let pα(n) denote the number of α-colored partitions of n. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real α>0, but when α>24 it involves several polar terms and a two-parameter family of exponential sums Ak(α)(n,m). For integral α, we prove multiplicativity and prime-power reduction formulas for these sums, expressing their local factors as classical or quadratically twisted Kloosterman sums. Together with explicit truncation and precision bounds, these formulas yield an efficient algorithm for computing pα(n) exactly. Our SageMath implementation computes the 1,113,767-digit integer p100(1010) in less than one hour. As an application, we use the algorithm to certify new Ramanujan-type congruences.
Related
- Internal congruences modulo powers of 2 for k-colored overpartitions into odd parts
- A uniform proof approach for congruences modulo 3 for partitions with k-colored odd parts
- Congruences modulo powers of $7$ for partitions with colored odd parts
- Nonexistence of Constant Congruences for the Cumulative Partition Function
- On Certain Congruences for Partition k-Tuples with $5$-Cores
BibTeX
@misc{villegasmorales2026efficient,
title = {{Efficient Computation and Congruences for Colored Partition Functions}},
author = {Jean Carlos Villegas-Morales},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32033},
eprint = {2609.32033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32033},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Efficient Computation and Congruences for Colored Partition Functions
AU - Jean Carlos Villegas-Morales
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32033
UR - https://arxiv.org/abs/2609.32033
AB - For a positive integer α, let p_α(n) denote the number of α-colored partitions of n. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real α>0, but when α>24 it involves several polar terms and a two-parameter family of exponential sums A_k^(α)(n,m). For integral α, we prove multiplicativity and prime-power reduction formulas for these sums, expressing their local factors as classical or quadratically twisted Kloosterman sums. Together with explicit truncation and precision bounds, these formulas yield an efficient algorithm for computing p_α(n) exactly. Our SageMath implementation computes the 1,113,767-digit integer p_100(10^10) in less than one hour. As an application, we use the algorithm to certify new Ramanujan-type congruences.
ER -