programming-examples/java/Data_Structures/Partitions.java

import java.util.*;

// https://en.wikipedia.org/wiki/Partition_(number_theory)
public class Partitions {

	public static boolean nextPartition(List<Integer> p) {
		int n = p.size();
		if (n <= 1)
			return false;
		int s = p.remove(n - 1) - 1;
		int i = n - 2;
		while (i > 0 && p.get(i).equals(p.get(i - 1))) {
			s += p.remove(i);
			--i;
		}
		p.set(i, p.get(i) + 1);
		while (s-- > 0) {
			p.add(1);
		}
		return true;
	}

	public static List<Integer> partitionByNumber(int n, long number) {
		List<Integer> p = new ArrayList<>();
		for (int x = n; x > 0; ) {
			int j = 1;
			while (true) {
				long cnt = partitionFunction(x)[x][j];
				if (number < cnt)
					break;
				number -= cnt;
				++j;
			}
			p.add(j);
			x -= j;
		}
		return p;
	}

	public static long numberByPartition(List<Integer> p) {
		long res = 0;
		int sum = 0;
		for (int x : p) {
			sum += x;
		}
		for (int cur : p) {
			for (int j = 0; j < cur; j++) {
				res += partitionFunction(sum)[sum][j];
			}
			sum -= cur;
		}
		return res;
	}

	public static void generateIncreasingPartitions(int[] p, int left, int last, int pos) {
		if (left == 0) {
			for (int i = 0; i < pos; i++)
				System.out.print(p[i] + " ");
			System.out.println();
			return;
		}
		for (p[pos] = last + 1; p[pos] <= left; p[pos]++)
			generateIncreasingPartitions(p, left - p[pos], p[pos], pos + 1);
	}

	public static long countPartitions(int n) {
		long[] p = new long[n + 1];
		p[0] = 1;
		for (int i = 1; i <= n; i++) {
			for (int j = i; j <= n; j++) {
				p[j] += p[j - i];
			}
		}
		return p[n];
	}

	public static long[][] partitionFunction(int n) {
		long[][] p = new long[n + 1][n + 1];
		p[0][0] = 1;
		for (int i = 1; i <= n; i++) {
			for (int j = 1; j <= i; j++) {
				p[i][j] = p[i - 1][j - 1] + p[i - j][j];
			}
		}
		return p;
	}

	public static long[][] partitionFunction2(int n) {
		long[][] p = new long[n + 1][n + 1];
		p[0][0] = 1;
		for (int i = 1; i <= n; i++) {
			for (int j = 1; j <= i; j++) {
				for (int k = 0; k <= j; k++) {
					p[i][j] += p[i - j][k];
				}
			}
		}
		return p;
	}

	// Usage example
	public static void main(String[] args) {
		System.out.println(7 == countPartitions(5));
		System.out.println(627 == countPartitions(20));
		System.out.println(5604 == countPartitions(30));
		System.out.println(204226 == countPartitions(50));
		System.out.println(190569292 == countPartitions(100));

		List<Integer> p = new ArrayList<>();
		Collections.addAll(p, 1, 1, 1, 1, 1);
		do {
			System.out.println(p);
		} while (nextPartition(p));

		int[] p1 = new int[8];
		generateIncreasingPartitions(p1, p1.length, 0, 0);

		List<Integer> list = partitionByNumber(5, 6);
		System.out.println(list);

		System.out.println(numberByPartition(list));
	}
}