Ayyer, A and Sinha, S (2024) Cores of Partitions in Rectangles. In: Electronic Journal of Combinatorics, 31 (1).

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Abstract
For a positive integer t â�¥ 2, the tcore of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of tcores of partitions contained in an r Ã� s rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves tcores and compute its asymptotics for large r, s. We then prove that the number of partitions inside the rectangle whose tcores are a fixed partition Ï� is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the tcore of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer n (Ann. Appl. Prob. 2023). In particular, we show that in the limit as r, s â�� â�� maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter Î± = (t  1)/2 and rate parameter Î² that depends on the aspect ratio. Mathematics Subject Classifications: 05A15, 05A16, 05A17, 60C05 Â© The authors.
Item Type:  Journal Article 

Publication:  Electronic Journal of Combinatorics 
Publisher:  Australian National University 
Additional Information:  The copyright for this article belongs to Australian National University. 
Department/Centre:  Division of Physical & Mathematical Sciences > Mathematics 
Date Deposited:  14 May 2024 10:28 
Last Modified:  14 May 2024 10:28 
URI:  https://eprints.iisc.ac.in/id/eprint/84432 
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