#### Alladi Ramakrishnan Hall

#### Webinar: Random t-cores and hook lengths in random partitions.

#### Shubham Sinha

##### UCSD

*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 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. This talk is based on the arxiv preprint arXiv:1911.03135.*

Zoom meeting ID: 839 6169 2184

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