Self-conjugate $t$-core partitions and applications

TL;DR

This paper establishes new criteria for identifying self-conjugate t-core partitions using hook length formulas and explores their applications to number theory, including formulas for Hurwitz class numbers.

Contribution

It introduces novel necessary and sufficient conditions for self-conjugate t-core partitions based on parts of the associated distinct odd partitions, expanding understanding of partition bijections.

Findings

New hook length formula for self-conjugate t-core partitions

Applications to subsets of natural numbers via the supernorm statistic

Derivation of a new formula for Hurwitz class numbers

Abstract

Partition theory abounds with bijections between different types of partitions. One of the most famous partition bijections maps each self-conjugate partition of a positive integer $n$ to a partition of $n$ into distinct odd parts, and vice versa. Here we prove new necessary and sufficient conditions for a self-conjugate partition to be $t$-core, in terms of only the parts of the corresponding partition into distinct odd parts, by proving a new hook length formula. Corollaries of these results include new applications of $t$-core self-conjugate partitions to subsets of the natural numbers, due to the recent investigation of a new partition statistic called the supernorm by the first author, Just, and Schneider, as well as many results on $t$-cores by Bringmann, Kane, Males, Ono, Raji, and others. We provide several examples of these applications, one of which gives a new formula for certain families of Hurwitz class numbers.