Analogues of Alder-Type Partition Inequalities for Fixed Perimeter Partitions

In a 2016 paper, Straub proved an analogue to Euler's celebrated partition identity for partitions with a fixed perimeter. Later, Fu and Tang provided both a refinement and generalization of Straub's analogue to $d$-distinct partitions. They also prove a related result to the first Rogers-Ramanujan identity by defining two new functions, $h_d(n)$ and $f_d(n)$ for a fixed perimeter $n$, that resemble the preexisitng $q_d$ and $Q_d$ functions. Motivated by generalizations of Alder's ex-conjecture, we further generalize the work done by Fu and Tang by introducing a new parameter $a$, similar to the work of Kang and Park. We observe the prevalence of binomial coefficients in our study of fixed perimeter partitions and use this to develop a more direct analogue to $Q_d$. Using combinatorial techniques, we find Alder-type partition inequalities in a fixed perimeter setting, specifically a reverse Alder-type inequality.