Partitions with fixed largest hook length

TL;DR

This paper investigates the distribution of integer partitions based on their largest hook length, extending classical partition identities and uncovering new analogues and congruences.

Contribution

It introduces new partition theorems related to largest hook length, generalizes existing identities, and connects these results with classical partition theory.

Findings

Refined Straub's analogue of Euler's Odd-Distinct theorem

Derived a generalization related to Alder's conjecture

Established an analogue of the Rogers-Ramanujan identity

Abstract

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence properties.