## Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations

##### Date

2017

##### Authors

Alanazi, Abdulaziz Mohammed

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##### Abstract

In this thesis, various partition functions with respect to `-regular overpartitions, a
special partition inequality and partition con gurations are studied.
We explore new combinatorial properties of overpartitions which are natural generalizations
of integer partitions. Building on recent work, we state general combinatorial
identities between standard partition, overpartition and `-regular partition
functions. We provide both generating function and bijective proofs.
We then establish an in nite set of Ramanujan-type congruences for the `-regular
overpartitions. This signi cantly extends the recent work of Shen which focused
solely on 3{regular overpartitions and 4{regular overpartitions. We also prove some
of the congruences for `-regular overpartition functions combinatorially.
We then provide a combinatorial proof of the inequality p(a)p(b) > p(a+b), where
p(n) is the partition function and a; b are positive integers satisfying a+b > 9, a > 1
and b > 1. This problem was posed by Bessenrodt and Ono who used the inequality
to study a maximal multiplicative property of an extended partition function.
Finally, we consider partition con gurations introduced recently by Andrews and
Deutsch in connection with the Stanley-Elder theorems. Using a variation of Stanley's
original technique, we give a combinatorial proof of the equality of the number
of times an integer k appears in all partitions and the number of partition con-
gurations of length k. Then we establish new generalizations of the Elder and
con guration theorems. We also consider a related result asserting the equality
of the number of 2k's in partitions and the number of unrepeated multiples of k,
providing a new proof and a generalization.

##### Description

A thesis submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, in ful lment of the requirements for
the degree of Doctor of Philosophy.
Johannesburg, 2017.

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##### Citation

Alanazi, Abdulaziz Mohammed (2017) Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/22738>