Johnson's bijections and their application to counting simultaneous core partitions

TL;DR

This paper explores Johnson's bijections to count and analyze simultaneous core partitions, providing new formulas for multi-core counts and the maximum size of certain self-conjugate core partitions.

Contribution

It extends Johnson's work by deriving formulas for the number of multi-core partitions with coprime parameters and determining the largest size of specific self-conjugate core partitions.

Findings

Derived an expression for the number of multi-core partitions with at least one pair of coprime parameters.

Evaluated the maximum size of self-conjugate (s,s+1,s+2)-core partitions.

Abstract

Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,\cdots, b_n)$-core partitions where $\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.