
The Limiting Distribution of the Hook Length of a Randomly Chosen Cell in a Random Young Diagram
L. R. Mutafchiev^{ab} ^{a} American University in Bulgaria
^{b} Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Abstract:
Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $\lambda$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $\lambda$ has a unique graphical representation, composed by $n$ nonoverlapping 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 the $n$ cells of the Young diagram of the partition $\lambda$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(\lambda,c)$ of the cell $c$ of a random partition $\lambda$. This twostep sampling procedure suggests a product probability measure, which assigns the probability $1/np(n)$ to each pair $(\lambda,c)$. With respect to this probability measure, we show that the random variable $\pi Z_n/\sqrt{6n}$ converges weakly, as $n\to\infty$, to a random variable whose probability density function equals $6y/\pi^2(e^y1)$ if $0<y<\infty$, and zero elsewhere. Our method of proof is based on Hayman's saddle point approach for admissible
power series.
Received: February 16, 2021 Revised: March 19, 2021 Accepted: September 27, 2021
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