The size of \(t\)-cores and hook lengths of random cells in random partitions. (English) Zbl 1515.60037

Ayyer, Arvind; Sinha, Shubham

Ann. Appl. Probab. 33, No. 1, 85-106 (2023).

Summary: Fix \(t\geq 2\). 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 dividing by \(\sqrt{n}\). As a consequence, we find that the size of the \(t\)-core is of the order of \(\sqrt{n}\) 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.

MSC:

60C05	Combinatorial probability
05A15	Exact enumeration problems, generating functions
05A17	Combinatorial aspects of partitions of integers
05E10	Combinatorial aspects of representation theory
11P82	Analytic theory of partitions

Keywords:

abacus; gamma distribution; hook length; \(t\)-core; \(t\)-quotient; uniformly random cell; uniformly random partition

Software:

SageMath

References:

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