<abstract xmlns="http://www.w3.org/1999/xhtml"> Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn. </abstract>

Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn.

Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

American University in Bulgaria 2700 Blagoevgrad, Bulgaria, and Institute of Mathematics and Informatics Bulgarian Academy of Sciences

Pure Mathematics and Applications

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

Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different...

Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

Published Online: Oct 07, 2015

Page range: 79 - 95

Received: Sep 08, 2014

DOI: https://doi.org/10.1515/puma-2015-0007

Keywords
integer partition, part size, random sampling, limiting distribution

© 2015

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

Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

Published Online: Oct 07, 2015

Page range: 79 - 95

Received: Sep 08, 2014

DOI: https://doi.org/10.1515/puma-2015-0007

Keywordsinteger partition, part size, random sampling, limiting distribution

© 2015

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

Keywords
integer partition, part size, random sampling, limiting distribution