Recursion: Java programming with integer partitioning to achieve __ programming

Integer Partitioning:

Positive integer n represents the sum of a series of positive integers:

N=n1+n2+n3...+nk (of which, n1>=n2>=n3...>=nk>=1,k>=1), p (n) is the number of different partitions of positive integer n, that is, the number of positive integers.

So in all the different divisions of positive integer n, the number of partitions with the maximum addends not greater than M is recorded as Q (n,m). Then the recursive relationship is as follows:

(1) q (n,1) =1,n>=1

M=1, that is, the maximum number can only be 1, the division of only one: n=1+1+1+.....+1;

(2) Q (n,m) =q (n,n), m>=n

Since the maximum addends is greater than or equal to the positive integer n, all the maximum addends can only be n;

(3) Q (n,n) =1+q (n,n-1)

When the maximum addends is n, there is only one case. The remaining is the number of addends when the maximum is n-1;

(4) Q (n,m) =q (n,m-1) +q (n-m,m), n>m>1

When the positive integer n is greater than the maximum addends m, Q (n,m-1) represents the number of partitions when the maximum addends is less than or equal to m-1, and then the number of partitions when the maximum addends is M is added. When the maximum addends is M, the maximum addends has been fixed, and the rest is the positive integer n minus the number of integers n-m the maximum addends m, that is, Q (n-m,m).

Program:

Import Java.util.Scanner;


The public class Intdivision {

	/**
	 * Positive integer n represents the sum of a series of positive integers:
	 * N=N1+N2+N3...+NK (where n1>=n2>=n3...>=nk>= 1,k>=1)
	 * p (N) is the number of different partitions of positive integer n, that is, the dividing number of positive integers
	 */
	int intdivision (int n,int m) {
		if n<1| | M<1) return 0;
		if (n==1| | M==1) return 1;
		if (n<m) return intdivision (n, n);
		if (n==m) return intdivision (n, m-1) +1;
		Return Intdivision (n, m-1) +intdivision (N-m, M);
	}
	public static void Main (string[] args) {
		Scanner in=new Scanner (system.in);
		System.out.print ("Enter a positive integer to be Divided");
		int N=in.nextint ();
		System.out.print ("Input Max Addends");
		int M=in.nextint ();
		Intdivision id=new intdivision ();
		
		System.out.println ("integer" +n+ "" + "in the case of the maximum Addends" +m+ "is the number of divisions:" +id.intdivision (n, m));

	}