Hook lengths and 3-cores

TL;DR

This paper proves a conjecture linking hook length formulas of partitions to the enumeration of 3-core partitions, establishing a precise zero-value correspondence between two combinatorial sequences.

Contribution

It confirms a conjecture connecting hook length-based formulas with 3-core partition counts, advancing understanding in partition combinatorics.

Findings

Proved the conjecture relating A000731(n)=0 and A033687(n)=0.

Established a new link between hook length formulas and 3-core partitions.

Enhanced combinatorial formulas for partition analysis.

Abstract

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture.