The asymptotic normality of $(s,s+1)$-cores with distinct parts

TL;DR

This paper proves that the size distribution of (s,s+1)-core partitions with distinct parts approaches a normal distribution as s grows large, confirming a conjecture in algebraic combinatorics.

Contribution

It provides a rigorous proof of the asymptotic normality of (s,s+1)-core partitions with distinct parts, using the Combinatorial Central Limit Theorem.

Findings

Distribution is approximately normal for large s

Confirms Zaleski's conjecture on asymptotic behavior

Uses mixing of normal distributions for proof

Abstract

Simultaneous core partitions are important objects in algebraic combinatorics. Recently there has been interest in studying the distribution of sizes among all $(s,t)$-cores for coprime $s$ and $t$. Zaleski (2017) gave strong evidence that when we restrict our attention to $(s,s+1)$-cores with distinct parts, the resulting distribution is approximately normal. We prove his conjecture by applying the Combinatorial Central Limit Theorem and mixing the resulting normal distributions.