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PreprintCC BY-NC-ND 4.0arXiv

Fixed-Perimeter Franklin Statistics and an Eventual Inequality

  • Marcus McCrea
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Abstract

Gray, Payne, Swisher, and Watson conjectured that, for fixed integers j ≥ 0 and k ≥ 2, the number FDj,k(n) of partitions of perimeter n having exactly j part sizes of multiplicity at least k is eventually at least the number FOj,k(n) having exactly j distinct occurring part sizes divisible by k. We derive bivariate generating functions for both statistics using the profile-word encoding of a partition. For k ≥ 3, coefficient extraction shows that fixing j changes the order of the dominant pole but not its location. The corresponding dominant singularities are positive real numbers ρk and σk, where ρk satisfies ρk+ρk2+⋯+ρkk=1 and σk satisfies σkk=(1−σk)k−1. We prove that ρk<σk for every k ≥ 3. Consequently, FDj,k(n)/FOj,k(n) → ∞ as n → ∞, proving the conjecture and yielding a strict eventual inequality for k ≥ 3. The case k=2 recovers the known exact identity.

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Cite

BibTeXRIS
BibTeX
@misc{mccrea2026fixedperimeter,
  title = {{Fixed-Perimeter Franklin Statistics and an Eventual Inequality}},
  author = {Marcus McCrea},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12005},
  eprint = {2609.12005},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12005},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Fixed-Perimeter Franklin Statistics and an Eventual Inequality
AU  - Marcus McCrea
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.12005
UR  - https://arxiv.org/abs/2609.12005
AB  - Gray, Payne, Swisher, and Watson conjectured that, for fixed integers j ≥ 0 and k ≥ 2, the number FD_j,k(n) of partitions of perimeter n having exactly j part sizes of multiplicity at least k is eventually at least the number FO_j,k(n) having exactly j distinct occurring part sizes divisible by k. We derive bivariate generating functions for both statistics using the profile-word encoding of a partition. For k ≥ 3, coefficient extraction shows that fixing j changes the order of the dominant pole but not its location. The corresponding dominant singularities are positive real numbers ρ_k and σ_k, where ρ_k satisfies ρ_k+ρ_k^2+⋯+ρ_k^k=1 and σ_k satisfies σ_k^k=(1−σ_k)^k−1. We prove that ρ_k<σ_k for every k ≥ 3. Consequently, FD_j,k(n)/FO_j,k(n) → ∞ as n → ∞, proving the conjecture and yielding a strict eventual inequality for k ≥ 3. The case k=2 recovers the known exact identity.
ER  -