Generalization of a few results in Integer Partitions

Abstract: 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 thenumber of parts', and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.