Large-scale Rook Placements

TL;DR

This paper investigates the asymptotic behavior of rook placements on growing Young diagrams defined by piecewise linear functions and reveals a limit shape phenomenon in the normalized cumulative X-ray of random permutations.

Contribution

It introduces a new framework for analyzing rook placements on large, shape-defined Young diagrams and establishes a limit shape result for the associated permutation X-ray.

Findings

Asymptotic formulas for rook placements on large Young diagrams

Proof of a limit shape phenomenon in the normalized cumulative X-ray

Connection between Young diagram growth and permutation limit shapes

Abstract

For each certain "nice" piecewise linear function $f:[0,1] \to [0,1]$, we consider a family of growing Young diagrams $\{\lambda(f,N)\}_{N=1}^{\infty}$ by enlarging the region under the graph of $f$. We compute asymptotic formulas for the number of rook placements of the shape $\lambda(f,N)$. We prove that the normalized cumulative X-ray of a uniformly random permutation, as the size of the permutation grows, exhibits a limit shape phenomenon.