Generalization of a few results in Integer Partitions

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to...

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Bibliographic Details
Published in:arXiv.org 2011-11
Main Authors: Manosij Ghosh Dastidar, Sourav Sen Gupta
Format: Article
Language:English
Subjects:
Integers
Partitions
Partitions (mathematics)
Theorems
Online Access:Get full text
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Summary:In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.
ISSN:2331-8422