Séminaire Lotharingien de Combinatoire, 84B.58 (2020), 11 pp.

Arvind Ayyer and Shubham Sinha

Random t-Cores and Hook Lengths in Random Partitions

Abstract. Fix t>=2. We first give an asymptotic formula for certain sums of the number of t-cores. We then use this result to compute the distribution of the size of the t-core of a uniformly random partition of an integer n. We show that this converges weakly to a gamma distribution after appropriate rescaling. As a consequence, we find that the size of the t-core is of the order of n^1/2 in expectation. We then apply this result to show that the probability that t divides the hook length of a uniformly random cell in a uniformly random partition equals 1/t in the limit. Finally, we extend this result to all modulo classes of t using abacus representations for cores and quotients.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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