PreprintarXiv licencearXiv
A perimeter analogue of Franklin's identity and an inequality related to the parity of parts
Abstract
We prove two conjectures regarding partition perimeter inequalities. It was conjectured by Gray, Payne, and Watson that Franklin's partition identity becomes an eventual inequality if one replaces the size of the partition with its perimeter. We prove this conjecture by asymptotic analysis of the corresponding generating functions. Gray, Payne, Swisher, and Watson also conjectured that there is a bias for partitions with fixed perimeter to have more odd parts than even parts. We prove this conjecture by deriving the corresponding generating functions and give a positive recurrence for the coefficients. In the case that there are an equal number of odd parts and even parts we provide a connection to peakless Motzkin paths.
Related
- Fixed-Perimeter Franklin Statistics and an Eventual Inequality
- Sharp comparison between the perimeter and its fractional analogue in two dimensions
- PERIMETER ROTATIONS OF TRIANGLES AND THE ERDŐS-DEBRUNNER INEQUALITY
- Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code
- A functional inequality related to Domar's uniform boundedness theorem
BibTeX
@misc{cuthbertson2026perimeter,
title = {{A perimeter analogue of Franklin's identity and an inequality related to the parity of parts}},
author = {Philip Cuthbertson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17389},
eprint = {2609.17389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17389},
}
RIS
TY - UNPB
TI - A perimeter analogue of Franklin's identity and an inequality related to the parity of parts
AU - Philip Cuthbertson
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17389
UR - https://arxiv.org/abs/2609.17389
AB - We prove two conjectures regarding partition perimeter inequalities. It was conjectured by Gray, Payne, and Watson that Franklin's partition identity becomes an eventual inequality if one replaces the size of the partition with its perimeter. We prove this conjecture by asymptotic analysis of the corresponding generating functions. Gray, Payne, Swisher, and Watson also conjectured that there is a bias for partitions with fixed perimeter to have more odd parts than even parts. We prove this conjecture by deriving the corresponding generating functions and give a positive recurrence for the coefficients. In the case that there are an equal number of odd parts and even parts we provide a connection to peakless Motzkin paths.
ER -