Skip to content
DraconiticResearch
Search Sources API
190,335 studies31 sourcesUpdated 43 min ago
PreprintCC BY 4.0arXiv

The two-block odd partition function

  • Mircea Merca
Open at sourcePDFDOICite

Abstract

We investigate arithmetic and combinatorial properties of the function a(n), which is the signed number of partitions of n into exactly two distinct part sizes, each occurring an odd number of times. We prove that a(n)≥ 0 for all n, establishing a positivity phenomenon for a signed partition function arising from a double Lambert series. Furthermore, we show that a(n) satisfies nontrivial divisibility properties modulo $3$. The results reveal unexpected arithmetic regularity in a family of partition functions defined by odd multiplicity constraints and restricted support.

Versions

Related

  1. Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for pod(n)
    Mircea Merca
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. On short averages of the Fibonacci partition function
    Sam Chow, Owen Jones
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Nonexistence of Constant Congruences for the Cumulative Partition Function
    Kathy Q. Ji
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. The range and omitted values of a certain sequence involving the partition function
    Kalyan Chakraborty, Alisha Kazi, Manoj Upreti
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  5. Geometric decomposition of the d-dimensional hard-sphere partition function
    Luke K. Davis
    arXivPreprintCC BY-NC-SA 4.0Full textMathematical physicsStatistical & nonlinear physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{merca2026twoblock,
  title = {{The two-block odd partition function}},
  author = {Mircea Merca},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33770},
  eprint = {2609.33770},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33770},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The two-block odd partition function
AU  - Mircea Merca
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33770
UR  - https://arxiv.org/abs/2609.33770
AB  - We investigate arithmetic and combinatorial properties of the function a(n), which is the signed number of partitions of n into exactly two distinct part sizes, each occurring an odd number of times. We prove that a(n)≥ 0 for all n, establishing a positivity phenomenon for a signed partition function arising from a double Lambert series. Furthermore, we show that a(n) satisfies nontrivial divisibility properties modulo $3$. The results reveal unexpected arithmetic regularity in a family of partition functions defined by odd multiplicity constraints and restricted support.
ER  -