Alder-type partition inequality at the general level

TL;DR

This paper proves a generalized Alder-type partition inequality for all levels and most parameters, extending known identities and removing previous restrictions on excluded parts.

Contribution

It establishes a broad, level-independent Alder-type inequality for partitions, removing the need for the exclusion of a specific part in the inequality.

Findings

Proves the inequality holds for all levels with finitely many exceptions.

Extends the second Rogers-Ramanujan identity to a more general setting.

Removes the restriction on excluding the part $d+3-a$ in the inequality.

Abstract

A Known Alder-type partition inequality of level $a$, which involves the second Rogers-Ramanujan identity when the level $a$ is 2, states that the number of partitions of $n$ into parts differing by at least $d$ with the smallest part being at least $a$ is greater than or equal to that of partitions of $n$ into parts congruent to $\pm a \pmod{d+3}$, excluding the part $d+3-a$. In this paper, we prove that for all values of $d$ with a finite number of exceptions, an arbitrary level $a$ Alder-type partition inequality holds without requiring the exclusion of the part $d+3-a$ in the latter partition.