The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

János Komlós,Gábor Tusnády,Emily Sergel

doi:10.37236/8770

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.

Highlights

A partition of n is a weakly decreasing sequence λ = (λ1 λ2 · · · λk > 0) whose parts sum to n, i.e., λ1 + λ2 + · · · + λk = n
The (French) Ferrers diagram of a partition λ is an arrangement of boxes which is left-justified and whose ith row from the bottom contains λi boxes
Ekhad and Doron Zeilberger [EZ15] determined the entire limit distribution obtained by fixing t − s, taking the size of a random (s, t)-core, normalizing, and letting s → ∞

Summary

Introduction

Ekhad and Doron Zeilberger [EZ15] determined the entire limit distribution obtained by fixing t − s, taking the size of a random (s, t)-core, normalizing, and letting s → ∞. Somewhat surprisingly these distributions are not normal and are not known to be associated with other combinatorial problems.

The Combinatorial Central Limit Theorem

Normality for a fixed number of parts

The distribution of the weights

Proof of Theorem 1

Computational Proofs of the Lemmas