Fixed-Perimeter Franklin Statistics and an Eventual Inequality

Abstract

Gray, Payne, Swisher, and Watson conjectured that, for fixed integers $j\ge 0$ and $k\ge 2$, the number $F{D}_{j,k}(n)$ of partitions of perimeter $n$ having exactly $j$ part sizes of multiplicity at least $k$ is eventually at least the number $F{O}_{j,k}(n)$ 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 $k\ge 3$, 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 ${\rho }_k$ and ${\sigma }_k$, where ${\rho }_k$ satisfies ${\rho }_k+{\rho }_k^2+\cdots +{\rho }_k^k=1$ and ${\sigma }_k$ satisfies ${\sigma }_k^k=(1-{\sigma }_k{)}^{k-1}$. We prove that ${\rho }_k<{\sigma }_k$ for every $k\ge 3$. Consequently, $F{D}_{j,k}(n)/F{O}_{j,k}(n)\to \infty$ as $n\to \infty$, proving the conjecture and yielding a strict eventual inequality for $k\ge 3$. The case $k=2$ recovers the known exact identity.