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 � 2, the t-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of t-cores of partitions contained in an r � 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 t-cores and compute its asymptotics for large r, s. We then prove that the number of partitions inside the rectangle whose t-cores are a fixed partition � is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the t-core 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 � � maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter α = (t - 1)/2 and rate parameter β that depends on the aspect ratio. Mathematics Subject Classifications: 05A15, 05A16, 05A17, 60C05 © 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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