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Limit theorems for integer partitions and their generalisations
Thesis (PhD)--Stellenbosch University, 2012.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2012
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| _version_ | 1867614083931439104 |
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| access_status_str | Open Access |
| author | Ralaivaosaona, Dimbinaina |
| author2 | Wagner, Stephan |
| author_browse | Ralaivaosaona, Dimbinaina Wagner, Stephan |
| author_facet | Wagner, Stephan Ralaivaosaona, Dimbinaina |
| author_sort | Ralaivaosaona, Dimbinaina |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2012. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/20019 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:46:23.902Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/20019 Limit theorems for integer partitions and their generalisations Ralaivaosaona, Dimbinaina Wagner, Stephan Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Mathematical partitions Prime partitions Prime numbers Gaussian distribution Poisson distribution Dissertations -- Mathematics Theses -- Mathematics Thesis (PhD)--Stellenbosch University, 2012. ENGLISH ABSTRACT: Various properties of integer partitions are studied in this work, in particular the number of summands, the number of ascents and the multiplicities of parts. We work on random partitions, where all partitions from a certain family are equally likely, and determine moments and limiting distributions of the different parameters. The thesis focuses on three main problems: the first of these problems is concerned with the length of prime partitions (i.e., partitions whose parts are all prime numbers), in particular restricted partitions (i.e., partitions where all parts are distinct). We prove a central limit theorem for this parameter and obtain very precise asymptotic formulas for the mean and variance. The second main focus is on the distribution of the number of parts of a given multiplicity, where we obtain a very interesting phase transition from a Gaussian distribution to a Poisson distribution and further to a degenerate distribution, not only in the classical case, but in the more general context of ⋋-partitions: partitions where all the summands have to be elements of a given sequence ⋋ of integers. Finally, we look into another phase transition from restricted to unrestricted partitions (and from Gaussian to Gumbel-distribution) as we study the number of summands in partitions with bounded multiplicities. AFRIKAANSE OPSOMMING: Verskillende eienskappe van heelgetal-partisies word in hierdie tesis bestudeer, in die besonder die aantal terme, die aantal stygings en die veelvoudighede van terme. Ons werk met stogastiese partisies, waar al die partisies in ’n sekere familie ewekansig is, en ons bepaal momente en limietverdelings van die verskillende parameters. Die teses fokusseer op drie hoofprobleme: die eerste van hierdie probleme gaan oor die lengte van priemgetal-partisies (d.w.s., partisies waar al die terme priemgetalle is), in die besonder beperkte partisies (d.w.s., partisies waar al die terme verskillend is). Ons bewys ’n sentrale limietstelling vir hierdie parameter en verkry baie presiese asimptotiese formules vir die gemiddelde en die variansie. Die tweede hooffokus is op die verdeling van die aantal terme van ’n gegewe veelvoudigheid, waar ons ’n baie interessante fase-oorgang van ’n normaalverdeling na ’n Poisson-verdeling en verder na ’n ontaarde verdeling verkry, nie net in die klassieke geval nie, maar ook in die meer algemene konteks van sogenaamde ⋋-partities: partisies waar al die terme elemente van ’n gegewe ry ⋋ van heelgetalle moet wees. Doctoral 2012-01-31T10:10:46Z 2012-03-30T10:34:19Z 2012-01-31T10:10:46Z 2012-03-30T10:34:19Z 2012-03 Thesis http://hdl.handle.net/10019.1/20019 en_ZA Stellenbosch University 81 p. application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Mathematical partitions Prime partitions Prime numbers Gaussian distribution Poisson distribution Dissertations -- Mathematics Theses -- Mathematics Ralaivaosaona, Dimbinaina Limit theorems for integer partitions and their generalisations |
| title | Limit theorems for integer partitions and their generalisations |
| title_full | Limit theorems for integer partitions and their generalisations |
| title_fullStr | Limit theorems for integer partitions and their generalisations |
| title_full_unstemmed | Limit theorems for integer partitions and their generalisations |
| title_short | Limit theorems for integer partitions and their generalisations |
| title_sort | limit theorems for integer partitions and their generalisations |
| topic | Mathematical partitions Prime partitions Prime numbers Gaussian distribution Poisson distribution Dissertations -- Mathematics Theses -- Mathematics |
| url | http://hdl.handle.net/10019.1/20019 |
| work_keys_str_mv | AT ralaivaosaonadimbinaina limittheoremsforintegerpartitionsandtheirgeneralisations |