Some Properties of Partitions of Positive Integer as the Sum of Distinct Numbers

A partition of a positive integer is defined as a way of writing as an unordered sum of one or more positive integers. After it was first proposed by Leonhard Euler, many mathematicians and computer scientists have been interested in studying it. They used many kinds of researching tools and methods and have obtained a series of achievements which are widely used in graph theory and combinatorial mathematics. Based on previous researches by others, this paper studies the distinct partitions of positive integer and gets the relation between the distinct partition and the even partition of the positive integer. At the same time, through studying the distinct odd partition and the distinct even partition of the positive integer, this paper gets a series of counting equations about them.