A combinatorial proof of the supper symmetric property of hook length

TL;DR

This paper provides a combinatorial proof demonstrating the super symmetric property of hook lengths in Young diagrams, showing uniform distribution of different hook types.

Contribution

It introduces a combinatorial proof for the super symmetric property of hook lengths, a novel approach in understanding Young diagram structures.

Findings

Hook lengths of different types appear uniformly in Young diagrams.

A combinatorial proof confirms the super symmetric property.

Uniform distribution of hook types is established.

Abstract

There are $k$ kinds of length $k$ hooks with different arm length. Actually, this $k$ kinds appear uniformly in Young diagrams of size $n$. The property ``appear uniformly'' is called super symmetric. We give a combinatorial proof of the supper symmetric property of hook length.