A multiset hook length formula and some applications

TL;DR

This paper introduces a multiset hook length formula for integer partitions using combinatorial methods, rederives existing formulas, and proves a related hook-content formula, with applications in combinatorics and topological vertex theory.

Contribution

It presents a new multiset hook length formula for partitions, unifies existing formulas, and establishes a multiset hook-content formula, advancing combinatorial and topological applications.

Findings

Rederived three known hook length formulas.

Established a new multiset hook length formula.

Proved a multiset hook-content formula.

Abstract

A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.