using System;
using System.Numerics;
using System.Collections.Generic;

public class Test
{
    // Helper functions for calculating values of (n choose k).
    // These are not optimally coded!
	// ----------------------------------------------------------------------
	protected static BigInteger factorial(int n) {
		BigInteger f = 1;
		while (n > 1) f *= n--;
		return f;
	}
	
	protected static int binomial(int n, int k) {
		if (k > n) return 0;
		return (int)(factorial(n) / (factorial(k) * factorial(n-k)));
	}
	
	// In the combinatorial number system, a combination {c_1, c_2, ..., c_k}
	// corresponds to the integer value obtained by adding (c_1 choose 1) +
	// (c_2 choose 2) + ... + (c_k choose k)
	// NOTE: combination values are assumed to start from zero, so
	// a combination like {1, 2, 3, 4, 5} will give a non-zero result
	// ----------------------------------------------------------------------
	public static int combination_2_index(int[] combo) {
		int ix = 0, i = 1;
		Array.Sort(combo);
		foreach (int c in combo) {
			if (c > 0) ix += binomial(c, i);
			i++;
		}
		return ix;
	}
	
	// The reverse of this process is a bit fiddly. See Wikipedia for an
	// explanation: https://e...content-available-to-author-only...a.org/wiki/Combinatorial_number_system
	// ----------------------------------------------------------------------
	public static int[] index_2_combination(int ix, int k) {
		List<int> combo_list = new List<int>();
		while (k >= 1) {
			int n = k - 1;
			if (ix == 0) {
				combo_list.Add(n);
				k--;
				continue;
			}
			int b = 0;
			while (true) {
			    // (Using a linear search here, but a binary search with
			    // precomputed binomial values would be faster)
				int b0 = b;
				b = binomial(n, k);
				if (b > ix || ix == 0) {
					ix -= b0;
					combo_list.Add(n-1);
					break;
				}
				n++;
			}
			k--;
		}
		int[] combo = combo_list.ToArray();
		Array.Sort(combo);
		return combo;
	}
	
	public static void Main()
	{
		int[][] test_combos = new int[][] {
		    new int[] {0,1,2,3,4,5},
		    new int[] {0,1,2,3,4,6},
		    new int[] {38,40,41,42,43,44},
		    new int[] {39,40,41,42,43,44}
		};
		int ix;
		for (int i=0; i<test_combos.Length; i++) {
			ix = combination_2_index(test_combos[i]);
			int[] combo = index_2_combination(ix, test_combos[i].Length);
			Console.WriteLine("{0} -> [{1}]", ix, string.Join(", ", combo));
		}
	}
}