CombinatorialEnumerations.java

package Combinatorics;
import java.util.Arrays;

//Enumeration of Cominatorial Sequences. Universal Method in 0(n^2)

public class CombinatorialEnumerations {

	// subclass and implement count() method
	public static abstract class AbstractEnumeration {

		// range of definition of sequence elements
		protected final int range;

		// length of generated sequences
		protected final int length;

		protected AbstractEnumeration(int range, int length) {
			this.range = range;
			this.length = length;
		}

		// returns number of combinatorial sequences starting with prefix
		// by contract only the last element of prefix can be invalid and in this case 0 must be returned
		protected abstract long count(int[] prefix);

		public int[] next(int[] sequence) {
			return fromNumber(toNumber(sequence) + 1);
		}

		public long totalCount() {
			return count(new int[0]);
		}

		public long toNumber(int[] sequence) {
			long res = 0;
			for (int i = 0; i < sequence.length; i++) {
				int[] prefix = Arrays.copyOf(sequence, i + 1);
				for (prefix[i] = 0; prefix[i] < sequence[i]; ++prefix[i]) {
					res += count(prefix);
				}
			}
			return res;
		}

		public int[] fromNumber(long number) {
			int[] sequence = new int[length];
			for (int i = 0; i < sequence.length; i++) {
				int[] prefix = Arrays.copyOf(sequence, i + 1);
				for (prefix[i] = 0; prefix[i] < range; ++prefix[i]) {
					long cur = count(prefix);
					if (number < cur) {
						break;
					}
					number -= cur;
				}
				sequence[i] = prefix[i];
			}
			return sequence;
		}

		public void enumerate() {
			System.out.println(getClass().getName());
			long total = totalCount();
			for (long i = 0; i < total; i++) {
				int[] p = fromNumber(i);
				System.out.println(i + " " + Arrays.toString(p));
				long j = toNumber(p);
				if (i != j)
					throw new RuntimeException();
			}
		}
	}

	public static class Arrangements extends AbstractEnumeration {

		public Arrangements(int n, int k) {
			super(n, k);
		}

		@Override
		protected long count(int[] prefix) {
			int size = prefix.length;

			// if the last element appears twice, then prefix is invalid and 0 must be returned
			for (int i = 0; i < size - 1; i++)
				if (prefix[size - 1] == prefix[i])
					return 0;

			long res = 1;
			for (int i = 0; i < length - size; i++)
				res *= range - size - i;
			return res;
		}
	}

	public static class Permutations extends Arrangements {
		public Permutations(int n) {
			super(n, n);
		}
	}

	public static class Combinations extends AbstractEnumeration {

		final long[][] binomial;

		public Combinations(int n, int k) {
			super(n, k);
			binomial = new long[n + 1][n + 1];
			// calculate binomial coefficients in advance
			for (int i = 0; i <= n; i++)
				for (int j = 0; j <= i; j++)
					binomial[i][j] = (j == 0) ? 1 : binomial[i - 1][j - 1] + binomial[i - 1][j];
		}

		@Override
		protected long count(int[] prefix) {
			int size = prefix.length;

			// if there is no combination with given prefix, 0 must be returned.
			// by contract only the last element can make prefix invalid.
			if (size >= 2 && prefix[size - 1] <= prefix[size - 2])
				return 0;

			// prefix is valid. return the number of combinations starting with prefix
			int last = size > 0 ? prefix[size - 1] : -1;
			return binomial[range - 1 - last][length - size];
		}
	}

	public static class CorrectBracketSequences extends AbstractEnumeration {

		final long[][] d;

		// sequenceLength must be a multiple of 2
		public CorrectBracketSequences(int sequenceLength) {
			super(2, sequenceLength);
			d = new long[sequenceLength + 1][sequenceLength / 2 + 1];
			// d[i][j] - number of bracket sequences of length i with balance j
			d[0][0] = 1;
			for (int i = 1; i <= sequenceLength; i++) {
				for (int balance = 0; balance <= sequenceLength / 2; balance++) {
					if (balance - 1 >= 0)
						d[i][balance] += d[i - 1][balance - 1];
					if (balance + 1 <= sequenceLength / 2)
						d[i][balance] += d[i - 1][balance + 1];
				}
			}
		}

		@Override
		protected long count(int[] prefix) {
			int size = prefix.length;

			int balance = 0;
			for (int cur : prefix)
				// 0 designates '('
				// 1 designates ')'
				balance += cur == 0 ? 1 : -1;

			if (balance < 0 || balance > length - size)
				return 0;

			return d[length - size][balance];
		}
	}

	public static class Partitions extends AbstractEnumeration {

		final long[][] pp;

		public Partitions(int value) {
			super(value + 1, value);
			long[][] p = new long[value + 1][value + 1];
			// p[i][j] - number of partitions of i with largest summand equal to j
			p[0][0] = 1;
			for (int i = 1; i <= value; i++)
				for (int j = 1; j <= i; j++)
					p[i][j] = p[i - 1][j - 1] + p[i - j][j];
			pp = new long[value + 1][value + 1];
			for (int i = 1; i <= value; i++)
				for (int j = 1; j <= value; j++)
					pp[i][j] = p[i][j] + pp[i][j - 1];
		}

		@Override
		protected long count(int[] prefix) {
			int size = prefix.length;
			int sum = 0;
			for (int e : prefix)
				sum += e;

			if (sum == range - 1)
				return 1;

			if (sum > range - 1 || size > 0 && prefix[size - 1] == 0 || size >= 2 && prefix[size - 1] > prefix[size - 2])
				return 0;

			int last = size > 0 ? prefix[size - 1] : range - 1;
			return pp[range - 1 - sum][last];
		}
	}

	public static void main(String[] args) {
		Permutations permutations = new Permutations(3);
		permutations.enumerate();

		Combinations combinations = new Combinations(4, 3);
		combinations.enumerate();

		Arrangements arrangements = new Arrangements(3, 2);
		arrangements.enumerate();

		CorrectBracketSequences correctBracketSequences = new CorrectBracketSequences(6);
		correctBracketSequences.enumerate();

		Partitions partitions = new Partitions(4);
		partitions.enumerate();
	}
}