{"id":{"repo_id":"stellenbosch","oai_identifier":"oai:scholar.sun.ac.za:10019.1/136012"},"canonical_url":"https://search.dev.ndltd.org/etd/stellenbosch/oai:scholar.sun.ac.za:10019.1/136012","repository":{"repo_id":"stellenbosch","name":"Stellenbosch University","base_url":"https://scholar.sun.ac.za/server/oai/request"},"display":{"title":"The Distribution of the Product of Parts in Integer Partitions","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Gikunda, Dennis Kinoti"],"institution":"Stellenbosch : Stellenbosch University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Ralaivaosaona, Dimbinaina"],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-03","date_published":"2026-03","updated_at":"2026-07-24T04:40:09Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholar.sun.ac.za/handle/10019.1/136012","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ralaivaosaona, Dimbinaina"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Stellenbosch University. 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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"]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["The Distribution of the Product of Parts in Integer Partitions"]}]}],"canonical_facts":{"dc:contributor.advisor":["Ralaivaosaona, Dimbinaina"],"dc:contributor.other":["Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences."],"dc:creator":["Gikunda, Dennis Kinoti"],"dc:date.accessioned":["2026-04-17T11:49:46Z"],"dc:date.available":["2026-04-17T11:49:46Z"],"dc:date.issued":["2026-03"],"dc:description":["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"],"dc:description.abstract":["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."],"dc:identifier.uri":["https://scholar.sun.ac.za/handle/10019.1/136012"],"dc:language.iso":["en"],"dc:publisher":["Stellenbosch : Stellenbosch University"],"dc:title":["The Distribution of the Product of Parts in Integer Partitions"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T04:40:09Z"}