Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions

TL;DR

This paper generalizes a conjecture related to partitions, showing that sums involving symmetric functions of hook lengths, contents, and parts are polynomial in n, extending identities inspired by Nekrasov and Okounkov.

Contribution

It proves that certain sums over partitions involving symmetric functions of hook lengths, contents, and parts are polynomial in n, generalizing previous conjectures.

Findings

Sums over partitions involving symmetric functions are polynomial in n.

Established polynomiality for functions of hook lengths, contents, and parts.

Extended identities related to Nekrasov and Okounkov.

Abstract

This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths of p, are polynomial functions of n. A similar result is obtained for symmetric functions of the contents and shifted parts of n.