
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 , we consider a family of growing Young diagrams by enlarging the region under the graph of . We compute asymptotic formulas for the number of rook placements of the shape . 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.
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Taxonomy
TopicsBayesian Methods and Mixture Models · Stochastic processes and statistical mechanics · Advanced Combinatorial Mathematics
