On Generalizations of a Conjecture of Kang and Park

Let $\Delta_d^{(a,-)}(n) = q_d^{(a)}(n) - Q_d^{(a,-)}(n)$ where $q_d^{(a)}(n)$ counts the number of partitions of $n$ into parts with difference at least $d$ and size at least $a$, and $Q_d^{(a,-)}(n)$ counts the number of partitions into parts $\equiv \pm a \pmod{d + 3}$ excluding the $d+3-a$ part. Motivated by generalizing a conjecture of Kang and Park, Duncan, Khunger, Swisher, and the second author conjectured that $\Delta_d^{(3,-)}(n)\geq 0$ for all $d\geq 1$ and $n\geq 1$ and were able to prove this when $d \geq 31$ is divisible by $3$. They were also able to conjecture an analog for higher values of $a$ that the modified difference function $\Delta_{d}^{(a,-,-)}(n) = q_{d}^{(a)}(n) - Q_{d}^{(a,-,-)}(n) \geq 0$ where $Q_{d}^{(a,-,-)}(n)$ counts the number of partitions into parts $\equiv \pm a \pmod{d + 3}$ excluding the $a$ and $d+3-a$ parts and proved it for infinitely many classes of $n$ and $d$. We prove that $\Delta_{d}^{(3,-)}(n) \geq 0$ for all but finitely many $d$. We also provide a proof of the generalized conjecture for all but finitely many $d$ for fixed $a$ and strengthen the results of Duncan, Khunger, Swisher, and the second author. We provide a conditional proof of a linear lower bound on $d$ for the generalized conjecture, which improves our unconditional result based on a conjectural modification of a recently proven conjecture of Alder. Using this modification, we obtain a strengthening of this generalization of Kang and Park's conjecture which remarkably allows $a$ as a part. Additionally, we provide asymptotic evidence that this strengthened conjecture holds.