On the largest part size and its multiplicity of a random integer partition

Ljuben Mutafchiev

doi:10.1515/puma-2015-0033

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On the largest part size and its multiplicity of a random integer partition

Pure Mathematics and Applications

Volume 28 (2019): Issue 1 (October 2019)

By: Ljuben Mutafchiev

Open Access

|Nov 2019

Abstract

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let L_n = L_n(λ) and M_n = M_n(λ) be the largest part size and its multiplicity, respectively. For large n, we focus on a comparison between the partition statistics L_n and L_nM_n. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of L_nM_n – L_n grows as fast as $\frac{1}{2} log n$ {1 \over 2}\log n. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.

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Articles in this issue

DOI: https://doi.org/10.1515/puma-2015-0033 | Journal eISSN: 1788-800X

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Language: English

Page range: 77 - 90

Submitted on: Aug 29, 2018

Accepted on: Sep 9, 2019

Published on: Nov 1, 2019

Published by: Corvinus University of Budapest

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Related subjects:

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© 2019 Ljuben Mutafchiev, published by Corvinus University of Budapest
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 28 (2019): Issue 1 (October 2019)