A Multi-Set Identity for Partitions

TL;DR

This paper proves a refined multiset identity relating cell arm and leg lengths in Ferrers diagrams, deepening understanding of their combinatorial structure.

Contribution

It establishes a new multiset identity connecting cell arm and leg lengths across all Ferrers diagrams with the same number of cells.

Findings

Proves a multiset identity for Ferrers diagram cells

Refines previous combinatorial identities

Enhances understanding of Ferrers diagram structure

Abstract

We prove that the multiset {(RightArmLength,LeftArmLength)} ranging over all cells of all Ferrers diagrams with n cells equals the multiset {(RightArmLength,LegLength)} ranging over all cells of all Ferrers diagrams with n cells, thereby refining a multi-set identity proved by C. Bessenrodt and by Bacher and L. Manivel. Added In revised version: Guo-Niu Han kindly pointed out to us that our main result is contained in reference [B.H] of the present article.