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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/20019 |
| Date | 03 1900 |
| Creators | Ralaivaosaona, Dimbinaina |
| Contributors | Wagner, S. G., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 81 p. |
| Rights | Stellenbosch University |
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