Hook lengths and shifted parts of partitions

TL;DR

This paper discusses identities related to partition hook lengths and shifted parts, proving a new symmetric function identity that contributes to the understanding of partition theory.

Contribution

It introduces a new symmetric function identity that advances the study of partition hook lengths and shifted parts, building on recent conjectures and previous formulas.

Findings

Proved a new symmetric function identity

Generalized conjectures on partition hook lengths

Connected identities to existing formulas in partition theory

Abstract

Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it.