Fixed-Perimeter Franklin Statistics and an Eventual Inequality

RESUMO

Gray, Payne, Swisher, and Watson conjectured that, for fixed integers $j \geq 0$ and $k \geq 2$, 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 $k \geq 3$, 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 $k \geq 3$. Consequently, $FD_{j,k}(n)/FO_{j,k}(n) \to \infty$ as $n \to \infty$, proving the conjecture and yielding a strict eventual inequality for $k \geq 3$. The case $k=2$ recovers the known exact identity.