The limiting distribution of the hook length of a randomly chosen cell in a random Young diagram

2019/06/16 by Ljuben Mutafchiev, Mutafchiev, Ljuben
Computer Science · Mathematics · #05A17 #11P82 #60C05 #60F05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1906.07169

openalex publication_date 2019/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p(n) be the number of all integer partitions of the positive integer n and let λ be a partition, selected uniformly at random from among all such p(n) partitions. It is known that each partition λ has a unique graphical representation, composed by n non-overlapping cells in the plane called Young diagram. As a second step of our sampling experiment, we select a cell c uniformly at random from among all n cells of the Young diagram of the partition λ. For large n, we study the asymptotic behavior of the hook length Z_n=Z_n(λ,c) of the cell c of a random partituion λ. This two-step sampling procedure suggests a product probability measure, which assigns the probability 1/np(n) to each pair (λ,c). With respect to this probability measure, we show that the random variable πZ_n/√(6n) converges weakly, as n→∞, to a random variable whose probability density function equals 6y/π² (e^y-1) if 0

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