THE SIZE OF t-CORES AND HOOK LENGTHS OF RANDOM CELLS IN RANDOM PARTITIONS
成果类型：
Article
署名作者：
Ayyer, Arvind; Sinha, Shubham
署名单位：
Indian Institute of Science (IISC) - Bangalore; University of California System; University of California San Diego
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/22-AAP1809
发表日期：
2023
页码：
85-106
关键词：
摘要：
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 dividing by root n. As a consequence, we find that the size of the t-core is of the order of root 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.