Toward resolving Kang and Park's generalization of the Alder-Andrews Theorem

TL;DR

This paper advances the understanding of a generalized Alder-Andrews Theorem by resolving nearly all remaining cases of a conjecture related to Rogers-Ramanujan identities using combinatorial and asymptotic methods.

Contribution

It extends Alfes et al.'s methods to nearly completely prove Kang and Park's conjecture on a level 2 Alder-Andrews type inequality.

Findings

Resolved nearly all remaining cases of Kang and Park's conjecture

Unified combinatorial and asymptotic approach for partition inequalities

Strengthened the connection between Rogers-Ramanujan identities and partition theory

Abstract

The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in 2008, and Alfes, Jameson, and Lemke Oliver in 2010. While Andrews and Yee utilized $q$-series and combinatorial methods, Alfes et al. proved the finite number of remaining cases using asymptotics originating with Meinardus together with high-performance computing. In 2020, Kang and Park conjectured a "level $2$" Alder-Andrews type partition inequality which relates to the second Rogers-Ramanujan identity. Duncan, Khunger, the second author, and Tamura proved Kang and Park's conjecture for all but finitely many cases using a combinatorial shift identity. Here, we generalize the methods of Alfes et al. to resolve nearly all of the remaining cases of Kang and Park's conjecture.