Inequalities for full rank differences of 2-marked Durfee symbols

TL;DR

This paper explores inequalities and identities related to full rank differences of 2-marked Durfee symbols, revealing their rarity and establishing an infinite family of non-trivial identities, contrasting with classical partition cases.

Contribution

It introduces new inequalities and identities for 2-marked Durfee symbols, highlighting their rarity and providing an infinite family of non-trivial identities.

Findings

Most inequalities for full rank differences are strict and almost always hold.

Non-trivial identities for full rank are rare, similar to Dyson's rank.

An infinite family of non-trivial identities for full rank differences is established.

Abstract

In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson's rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, contrasting the partition theoretic case.