Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for pod(n)
Abstract
The two-block odd partition function a(n) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the 2-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation a(2km)=a(m)+(2k−1−1)σ(m), where m is odd, k≥1, and σ(m) is the sum-of-divisors function. Combined with a previously established arithmetic formula for a(n), this relation yields congruences along the dyadic progressions 2k(4n+3), as well as the families a (2α3β(12n+11))≡0pmod3 and a (2α(18n+15))≡0pmod3. We also use a factorization of the generating function of a(n) involving pod(n), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving pod(n) are congruent to a(n) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo 4. As special cases, we obtain infinite families of weighted congruences modulo 3, 5, and 13.
Related
- Nonexistence of Constant Congruences for the Cumulative Partition Function
- Congruences for the Andrews--Uncu partition function ℰ𝒪u(n)
- On the weighted weak-type constant for the dyadic square function
- On Certain Congruences for Partition k-Tuples with $5$-Cores
- Congruences for Overcolored Partition k-tuples Restricted by Parity of the Parts
BibTeX
@misc{merca2026dyadic,
title = {{Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for $pod(n)$}},
author = {Mircea Merca},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05020},
eprint = {2610.05020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05020},
}
RIS
TY - UNPB
TI - Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for pod(n)
AU - Mircea Merca
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05020
UR - https://arxiv.org/abs/2610.05020
AB - The two-block odd partition function a(n) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the 2-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation a(2^km)=a(m)+(2^k−1−1)σ(m), where m is odd, k≥1, and σ(m) is the sum-of-divisors function. Combined with a previously established arithmetic formula for a(n), this relation yields congruences along the dyadic progressions 2^k(4n+3), as well as the families a (2^α3^β(12n+11))≡0pmod3 and a (2^α(18n+15))≡0pmod3. We also use a factorization of the generating function of a(n) involving pod(n), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving pod(n) are congruent to a(n) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo 4. As special cases, we obtain infinite families of weighted congruences modulo 3, 5, and 13.
ER -