Generalizations of Alder's Conjecture via a Conjecture of Kang and Park

Abstract

Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical physics. Here, we focus on partitions with gap conditions and partitions with parts coming from fixed residue classes. Let ${\Delta }_d^{(a,b)}(n)=q_d^{(a)}(n)-Q_d^{(b)}(n)$ 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 $Q_d^{(b)}(n)$ counts the number of partitions into parts $\equiv \pm b(\mathrm{mod}d+3)$. In 1956, Alder conjectured that ${\Delta }_d^{(1,1)}(n)\ge 0$ for all positive $n$ and $d$. This conjecture was proved partially by Andrews in 1971, by Yee in 2008, and was fully resolved by Alfes, Jameson and Lemke Oliver in 2011. Alder's conjecture generalizes several well-known partition identities, including Euler's theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts, as well as the first of the famous Rogers-Ramanujan identities. In 2020, Kang and Park constructed an extension of Alder's conjecture which relates to the second Rogers-Ramanujan identity by considering ${\Delta }_d^{(a,b,-)}(n)=q_d^{(a)}(n)-Q_d^{(b,-)}(n)$ where $Q_d^{(b,-)}(n)$ counts the number of partitions into parts $\equiv \pm b(\mathrm{mod}d+3)$ excluding the $d+3-b$ part. Kang and Park conjectured that ${\Delta }_d^{(2,2,-)}(n)\ge 0$ for all $d\ge 1$ and $n\ge 0$, and proved this for $d=2^r-2$ and $n$ even. We prove Kang and Park's conjecture for all but finitely many $d$. Toward proving the remaining cases, we adapt work of Alfes, Jameson and Lemke Oliver to generate asymptotics for the related functions. Finally, we present a more generalized conjecture for higher $a=b$ and prove it for infinite classes of $n$ and $d$.

24 pages