In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. If order matters, the sum becomes a composition. For example, 4 can be partitioned in five distinct ways:

4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while 1 + 2 + 1 and 1 + 1 + 2 are the same partition as 2 + 1 + 1.

If we have a partition of n, we can reduce it to a partition of n-1 in a canonical way by subtracting one from the smallest item in the partition. E.g. 1+2+3 => 2+3, 2+4 => 1+4. This algorithm reverses the process: for each partition p of n-1, it finds the partitions of n that would be reduced to p by this process. Therefore, each partition of n is output exactly once, at the step when the partition of n-1 to which it reduces is considered.

public class Partition { 
    public static void partition(int n) {
        partition(n, n, "");
    }
    public static void partition(int n, int max, String answer) {
        if (n == 0) {
            System.out.println(answer.substring(1));
            return;
        }
        for (int i = Math.min(max, n); i >= 1; i--) {
            partition(n-i, i, answer + "+" + i);
        }
    }
    public static void main(String[] args) { 
        int N = 4;
        partition(N);
    }
}

The purpose of the substring() function on line number 7 is to remove the ‘+’ sign at beginning of each line of output. As usual, the main function has purposely been kept to minimum and its expected that students will write appropriate input/output statements as per the requirement.