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The Distribution of the Product of Parts in Integer Partitions
Thesis (PhD)--Stellenbosch University, 2026.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : Stellenbosch University
2026
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| _version_ | 1867613859766861824 |
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| access_status_str | Open Access |
| author | Gikunda, Dennis Kinoti |
| author2 | Ralaivaosaona, Dimbinaina |
| author_browse | Gikunda, Dennis Kinoti Ralaivaosaona, Dimbinaina |
| author_facet | Ralaivaosaona, Dimbinaina Gikunda, Dennis Kinoti |
| author_sort | Gikunda, Dennis Kinoti |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2026. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/136012 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:42:50.594Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2026 |
| publishDateRange | 2026 |
| publishDateSort | 2026 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/136012 The Distribution of the Product of Parts in Integer Partitions Gikunda, Dennis Kinoti Ralaivaosaona, Dimbinaina Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Thesis (PhD)--Stellenbosch University, 2026. Gikunda, D. K. 2026. The Distribution of the Product of Parts in Integer Partitions. Unpublished doctoral dissertation. Stellenbosch: Stellenbosch University [online]. Available: https://scholar.sun.ac.za/items/48a9d3a1-2293-42d5-b5d0-08918547b69f Given a positive integer n, a partition L = (ℓ1, ℓ2, ℓ3, . . .) is a sequence of non-decreasing positive integers whose sum is n. The norm N(L) of a partition L is defined as the product of its parts. Consider a random partition L of n sampled uniformly from all partitions of n. It was recently proved by Bridges and Craig that N(L) lacks a non-trivial limiting distribution as n → ∞. We believe this is because the norm is, in a sense, a multiplicative statistic on partitions. Hence, in this work, we study instead the logarithm of the norm (or log-norm) log N(L). We prove that the log-norm of a random partition of n converges to a continuous limiting distribution as n → ∞. We extend this result to the case of Λ-partitions, where the parts of the partitions are restricted to elements of a fixed sequence of positive integers Λ. Under a general analytic framework due to Meinardus, we are also able to show the existence of a limiting distribution for the log-norm of a random Λ-partition. Notable examples include the cases of square partitions and prime partitions. Furthermore, we consider a non-uniform measure on the set of partitions of n, defined using a Vershik-style multiplicative weight. In this setting, we find that the limiting behavior of the distribution of the log-norm undergoes a phase transition: it changes from non-Gaussian to Gaussian. Finally, for the case of restricted partitions, where parts are not allowed to repeat, we prove that the distribution is almost always Gaussian. Doctoral 2026-04-17T11:49:46Z 2026-04-17T11:49:46Z 2026-03 Thesis https://scholar.sun.ac.za/handle/10019.1/136012 en Stellenbosch University 100 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Gikunda, Dennis Kinoti The Distribution of the Product of Parts in Integer Partitions |
| title | The Distribution of the Product of Parts in Integer Partitions |
| title_full | The Distribution of the Product of Parts in Integer Partitions |
| title_fullStr | The Distribution of the Product of Parts in Integer Partitions |
| title_full_unstemmed | The Distribution of the Product of Parts in Integer Partitions |
| title_short | The Distribution of the Product of Parts in Integer Partitions |
| title_sort | distribution of the product of parts in integer partitions |
| url | https://scholar.sun.ac.za/handle/10019.1/136012 |
| work_keys_str_mv | AT gikundadenniskinoti thedistributionoftheproductofpartsinintegerpartitions AT gikundadenniskinoti distributionoftheproductofpartsinintegerpartitions |