The size of $t$cores and hook lengths of random cells in random partitions
Abstract
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 appropriate rescaling. 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 continual Young diagrams and abacus representations for cores and quotients.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.03135
 Bibcode:
 2019arXiv191103135A
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 Mathematics  Number Theory;
 60B10;
 60C05;
 05A15;
 05A17;
 05E10;
 11P82
 EPrint:
 31 pages, 5 figures