Research Of Polygons With Integer Sides Based On The Partitions Of Positive Integer

The combinatorial mathematics is an active branch in the modern math field, while Partition of Positive Integer is one of the important question in number theory, combinatorial mathematics, graph theory and the application research. The Partition of Positive Integer is initially proposed by Leibniz in1699, which has also been referred to in the unpublished drafts of his. After many exquisite and important Partition Theorem has been proved by Euler(1799-1871), it has developed to be a relatively complete Partitions Theory. With the constant improvement of Partitions Theory.of positive integer and the widely application of its results, attracts many scholars in-depth study. Soon afterwards, scholar H. Jordan, G.E. Adrews, Linyan Xing and many others deeply studied the counting problem about Triangles with integer sides、Trapezoid with integer sides produced by the combination of Partition of Positive Integer and Geometric. On this basis, the paper concentrates on studying the counting problem to the number of the k polygon with integer sides separated by the perimeter of Positive Integer n.The main work of this paper includes the following several aspects:(1) Through the largest triangle theorem of Triangles with integer sides and the enumeration method, gives the counting formula of a new Triangles with integer sides, triangle, isosceles triangle with integer sides and quadrilateral with integer sides.(2) According to the order connection problem of each side of the polygon with integer sides, studied the problem of the Circle Permutation and Round Permutation and provided the counting formula to calculate any trimming the edges of the polygon can formφ(S) different whole polygon using Mobius inversion formula.(3)By solving the positive integer solution of the Indeterminate Equation x1+x2+…+xk=n,x1≤x2≤…≤xk,x1+x2+…+xtk-1>xk, we can arrive at the length of k polygon with integer sides. We can use the set Si={n1·x1,n2·x2,…,n1·x1}(notice: n1+n2+…+nk=k,nt·x1+n2·2+…+n1·x1=n,x1<x2<…<x1,i=p1,p2,…,p1) to represent the Indeterminate Equation the solution set, and giving the counting formula of the number of the Perimeter of the positive integer n and being partitioned into k polygons.