An involution for symmetry of hook length and part length of partitions

TL;DR

This paper introduces an involution on pointed partitions that swaps hook length and part length, providing a bijective proof of a recent combinatorial result.

Contribution

It constructs a new involution on pointed partitions that exchanges hook length and part length, offering a novel bijective proof.

Findings

Established an involution exchanging hook length and part length

Provided a bijective proof of Bessenrodt and Han's result

Enhanced understanding of partition symmetry properties

Abstract

A {\em pointed partition} of $n$ is a pair $(\lambda, v)$ where $\lambda\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a bijective proof of a recent result of Bessenrodt and Han.