<abstract xmlns="http://www.w3.org/1999/xhtml">Let λ be a partition of the positive integer <italic>n</italic> chosen uniformly at random among all such partitions. Let <italic>Ln</italic> = <italic>L</italic><italic>n</italic>(λ) and <italic>M</italic><italic>n</italic> = <italic>M</italic><italic>n</italic>(λ) be the largest part size and its multiplicity, respectively. For large <italic>n</italic>, we focus on a comparison between the partition statistics <italic>L</italic><italic>n</italic> and <italic>L</italic><italic>n</italic><italic>M</italic><italic>n</italic>. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of <italic>L</italic><italic>n</italic><italic>M</italic><italic>n</italic> – <italic>L</italic><italic>n</italic> grows as fast as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2015-0033_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>log</mo><mi>n</mi></mrow></math><tex-math>{1 \over 2}\log n</tex-math></alternatives></inline-formula>. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.</abstract>

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = Ln(λ) and Mn = Mn(λ) be the largest part size and its multiplicity, respectively. For large n, we focus on a comparison between the partition statistics Ln and LnMn. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of LnMn – Ln grows as fast as 12logn{1 \over 2}\log n. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.

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

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"On the largest part size and its multiplicity of a random integer partition"}

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = Ln(λ) and Mn = Mn(λ) be the largest part size and...

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

Published Online: Nov 01, 2019

Page range: 77 - 90

Received: Aug 29, 2018

Accepted: Sep 09, 2019

DOI: https://doi.org/10.1515/puma-2015-0033

Keywords
integer partition, part size, multiplicity, expectation, asymptotic behavior

© 2019 Ljuben Mutafchiev, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Nov 01, 2019

Page range: 77 - 90

Received: Aug 29, 2018

Accepted: Sep 09, 2019

DOI: https://doi.org/10.1515/puma-2015-0033

Keywordsinteger partition, part size, multiplicity, expectation, asymptotic behavior

© 2019 Ljuben Mutafchiev, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
integer partition, part size, multiplicity, expectation, asymptotic behavior