Count the number of ways that binary entropy can be at least 7 in a file of 256 characters chosen from 256 symbols; see https://math.stackexchange.com/questions/2858152.

Question2858152.java

import java.math.BigInteger;

public class Question2858152 {
	final static int ncharacters = 256;
	final static int nsymbols = 256;
	
	final static int max = Math.max (nsymbols,ncharacters) + 1;
	
	final static BigInteger [] integers = new BigInteger [max];
	final static BigInteger [] powers = new BigInteger [max];
	final static BigInteger [] factorials = new BigInteger [max];
	
	static BigInteger maxEntropy = BigInteger.TWO.pow (nsymbols);
	
	static BigInteger permutationCount = BigInteger.ZERO;
	static BigInteger combinationCount = BigInteger.ZERO;
	static BigInteger fileCount = BigInteger.ZERO;

	public static void main (String [] args) {
		factorials [0] = BigInteger.ONE;
		for (int i = 1;i < max;i++) {
			integers [i] = BigInteger.valueOf (i);
			powers [i] = integers [i].pow (i);
			factorials [i] = factorials [i - 1].multiply (integers [i]);
		}
		
		recurse (BigInteger.ONE,ncharacters,factorials [nsymbols],factorials [ncharacters],nsymbols,ncharacters);

		System.out.println (combinationCount + " combinations");
		System.out.println (permutationCount + " permutations");
		System.out.println (fileCount + " files");
		int precision = 100000000;
		BigInteger nfiles = integers [nsymbols].pow (ncharacters);
		System.out.println (fileCount.multiply (BigInteger.valueOf (precision)).add (nfiles.divide (BigInteger.TWO)).divide (nfiles).doubleValue () / precision);
	}
	
	static long time = System.currentTimeMillis ();
	
	static void recurse (BigInteger entropy,int frequency,BigInteger symbolMultiplicity,BigInteger characterMultiplicity,int symbolsLeft,int charactersLeft) {
		if (frequency == 1) {
			symbolMultiplicity = symbolMultiplicity.divide (factorials [charactersLeft]).divide (factorials [symbolsLeft - charactersLeft]);
			combinationCount = combinationCount.add (BigInteger.ONE);
			permutationCount = permutationCount.add (symbolMultiplicity);
			fileCount = fileCount.add (symbolMultiplicity.multiply (characterMultiplicity));
		}
		else
			for (int f = 0;charactersLeft >= 0;f++,charactersLeft -= frequency) {
				if (entropy.compareTo (maxEntropy) <= 0)
					recurse (entropy,frequency - 1,symbolMultiplicity.divide (factorials [f]),characterMultiplicity,symbolsLeft - f,charactersLeft);
				else
					break;
				if (f == 0 && symbolsLeft == nsymbols) {
					long now = System.currentTimeMillis ();
					System.out.println ("done with maximal frequency " + frequency + " (" + (now - time) / 1000 + " seconds)");
					time = now;
				}
				entropy = entropy.multiply (powers [frequency]);
				characterMultiplicity = characterMultiplicity.divide (factorials [frequency]);
			}
	}
}