A Positive Integer Spin-off Applications

A partition of a positive integer n is representation of n as an unordered sum of one or more positive integers. The number of different partitions of the positive integer n is called the partition number of n. The partition of the positive integer is an important issue in Combinatorics, Graph theory and Number theory. G.W.Leibniz is the first mathematician to study it, and then Euler expanded it to the complete theory of partitions.In this thesis, we studied some partitions of positive integer with restrained condition by combinatorial method and Ferrers graph of partition. In chapter 3, the partitions with some continuous odd or even part are discussed. A sufficient and necessary condition of the positive integer n, which can be represented as a sum of some continuous even or odd numbers is given. The partition numbers of these two kinds of partitions are also obtained. These consequences are used for research the equation x²-y² = n. The condition of the equation existence solution and number of solution are given.In chapter 4, we show the counting formula by primary method to convert O(n ,m) and e(n ,m) with finite O(n ,2) and e(n ,2), respectively. Thus we could calculate the value of O(n ,m) and e(n ,m) . And we also discussed a recursive counting method for the number of partition with distinct odd and even part, respectively. Where, O(n ,m) be the number of unordered partitions of an integer n into m odd positive integers, and e(n ,m) be the number of unordered partitions of a positive integer n into m even parts. In chapter 5, the question about m-partitions it is an extensive definition of the unordered partitions of positive integer n is studied. We give the generating function for p_k(n, m), the number of the m-partition of n with k parts. And we also show a relation for p_k(n, m) and Q(n ,k), the number of partitions of n into k distinct parts. Meanwhile, we get recursive relation for p_k(n, m). In addition, we discuss the application of the partitions in the number of positive integral solution of Diophantine equation x₁+2x₂+…+kx_k = n.In chapter 6, the partitions with three parts of positive integer and the triangle...