Task: Latin Squares in reduced form



C#: using System;
using System.Collections.Generic;
using System.Linq;

namespace LatinSquares {
    using matrix = List<List<int>>;

    class Program {
        static void Swap<T>(ref T a, ref T b) {
            var t = a;
            a = b;
            b = t;
        }

        static matrix DList(int n, int start) {
            start--; 
            var a = Enumerable.Range(0, n).ToArray();
            a[start] = a[0];
            a[0] = start;
            Array.Sort(a, 1, a.Length - 1);
            var first = a[1];
            
            matrix r = new matrix();
            void recurse(int last) {
                if (last == first) {
                    
                    
                    for (int j = 1; j < a.Length; j++) {
                        var v = a[j];
                        if (j == v) {
                            return; 
                        }
                    }
                    
                    var b = a.Select(v => v + 1).ToArray();
                    r.Add(b.ToList());
                    return;
                }
                for (int i = last; i >= 1; i--) {
                    Swap(ref a[i], ref a[last]);
                    recurse(last - 1);
                    Swap(ref a[i], ref a[last]);
                }
            }
            recurse(n - 1);
            return r;
        }

        static ulong ReducedLatinSquares(int n, bool echo) {
            if (n <= 0) {
                if (echo) {
                    Console.WriteLine("[]\n");
                }
                return 0;
            } else if (n == 1) {
                if (echo) {
                    Console.WriteLine("[1]\n");
                }
                return 1;
            }

            matrix rlatin = new matrix();
            for (int i = 0; i < n; i++) {
                rlatin.Add(new List<int>());
                for (int j = 0; j < n; j++) {
                    rlatin[i].Add(0);
                }
            }
            
            for (int j = 0; j < n; j++) {
                rlatin[0][j] = j + 1;
            }

            ulong count = 0;
            void recurse(int i) {
                var rows = DList(n, i);

                for (int r = 0; r < rows.Count; r++) {
                    rlatin[i - 1] = rows[r];
                    for (int k = 0; k < i - 1; k++) {
                        for (int j = 1; j < n; j++) {
                            if (rlatin[k][j] == rlatin[i - 1][j]) {
                                if (r < rows.Count - 1) {
                                    goto outer;
                                }
                                if (i > 2) {
                                    return;
                                }
                            }
                        }
                    }
                    if (i < n) {
                        recurse(i + 1);
                    } else {
                        count++;
                        if (echo) {
                            PrintSquare(rlatin, n);
                        }
                    }
                outer: { }
                }
            }

            
            recurse(2);
            return count;
        }

        static void PrintSquare(matrix latin, int n) {
            foreach (var row in latin) {
                var it = row.GetEnumerator();
                Console.Write("[");
                if (it.MoveNext()) {
                    Console.Write(it.Current);
                }
                while (it.MoveNext()) {
                    Console.Write(", {0}", it.Current);
                }
                Console.WriteLine("]");
            }
            Console.WriteLine();
        }

        static ulong Factorial(ulong n) {
            if (n <= 0) {
                return 1;
            }
            ulong prod = 1;
            for (ulong i = 2; i < n + 1; i++) {
                prod *= i;
            }
            return prod;
        }

        static void Main() {
            Console.WriteLine("The four reduced latin squares of order 4 are:\n");
            ReducedLatinSquares(4, true);

            Console.WriteLine("The size of the set of reduced latin squares for the following orders");
            Console.WriteLine("and hence the total number of latin squares of these orders are:\n");
            for (int n = 1; n < 7; n++) {
                ulong nu = (ulong)n;

                var size = ReducedLatinSquares(n, false);
                var f = Factorial(nu - 1);
                f *= f * nu * size;
                Console.WriteLine("Order {0}: Size {1} x {2}! x {3}! => Total {4}", n, size, n, n - 1, f);
            }
        }
    }
}



Java: import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class LatinSquaresInReducedForm {

    public static void main(String[] args) {
        System.out.printf("Reduced latin squares of order 4:%n");
        for ( LatinSquare square : getReducedLatinSquares(4) ) {
            System.out.printf("%s%n", square);
        }
        
        System.out.printf("Compute the number of latin squares from count of reduced latin squares:%n(Reduced Latin Square Count) * n! * (n-1)! = Latin Square Count%n");
        for ( int n = 1 ; n <= 6 ; n++ ) {
            List<LatinSquare> list = getReducedLatinSquares(n);
            System.out.printf("Size = %d, %d * %d * %d = %,d%n", n, list.size(), fact(n), fact(n-1), list.size()*fact(n)*fact(n-1));
        }
    }
    
    private static long fact(int n) {
        if ( n == 0 ) {
            return 1;
        }
        int prod = 1;
        for ( int i = 1 ; i <= n ; i++ ) {
            prod *= i;
        }
        return prod;
    }
    
    private static List<LatinSquare> getReducedLatinSquares(int n) {
        List<LatinSquare> squares = new ArrayList<>();
        
        squares.add(new LatinSquare(n));
        PermutationGenerator permGen = new PermutationGenerator(n);
        for ( int fillRow = 1 ; fillRow < n ; fillRow++ ) {
            List<LatinSquare> squaresNext = new ArrayList<>();
            for ( LatinSquare square : squares ) {
                while ( permGen.hasMore() ) {
                    int[] perm = permGen.getNext();
                    
                    
                    if ( (perm[0]+1) != (fillRow+1) ) {
                        continue;
                    }
                    
                    
                    boolean permOk = true;
                    done:
                    for ( int row = 0 ; row < fillRow ; row++ ) {
                        for ( int col = 0 ; col < n ; col++ ) {
                            if ( square.get(row, col) == (perm[col]+1) ) {
                                permOk = false;
                                break done;
                            }
                        }
                    }
                    if ( permOk ) {
                        LatinSquare newSquare = new LatinSquare(square);
                        for ( int col = 0 ; col < n ; col++ ) {
                            newSquare.set(fillRow, col, perm[col]+1);
                        }
                        squaresNext.add(newSquare);
                    }
                }
                permGen.reset();
            }
            squares = squaresNext;
        }
        
        return squares;
    }
    
    @SuppressWarnings("unused")
    private static int[] display(int[] in) {
        int [] out = new int[in.length];
        for ( int i = 0 ; i < in.length ; i++ ) {
            out[i] = in[i] + 1;
        }
        return out;
    }
    
    private static class LatinSquare {
        
        int[][] square;
        int size;
        
        public LatinSquare(int n) {
            square = new int[n][n];
            size = n;
            for ( int col = 0 ; col < n ; col++ ) {
                set(0, col, col + 1);
            }
        }
        
        public LatinSquare(LatinSquare ls) {
            int n = ls.size;
            square = new int[n][n];
            size = n;
            for ( int row = 0 ; row < n ; row++ ) {
                for ( int col = 0 ; col < n ; col++ ) {
                    set(row, col, ls.get(row, col));
                }
            }
        }
        
        public void set(int row, int col, int value) {
            square[row][col] = value;
        }

        public int get(int row, int col) {
            return square[row][col];
        }

        @Override
        public String toString() {
            StringBuilder sb = new StringBuilder();
            for ( int row = 0 ; row < size ; row++ ) {
                sb.append(Arrays.toString(square[row]));
                sb.append("\n");
            }
            return sb.toString();
        }
        
        
    }

    private static class PermutationGenerator {

        private int[] a;
        private BigInteger numLeft;
        private BigInteger total;

        public PermutationGenerator (int n) {
            if (n < 1) {
                throw new IllegalArgumentException ("Min 1");
            }
            a = new int[n];
            total = getFactorial(n);
            reset();
        }

        private void reset () {
            for ( int i = 0 ; i < a.length ; i++ ) {
                a[i] = i;
            }
            numLeft = new BigInteger(total.toString());
        }

        public boolean hasMore() {
            return numLeft.compareTo(BigInteger.ZERO) == 1;
        }

        private static BigInteger getFactorial (int n) {
            BigInteger fact = BigInteger.ONE;
            for ( int i = n ; i > 1 ; i-- ) {
                fact = fact.multiply(new BigInteger(Integer.toString(i)));
            }
            return fact;
        }

        
        public int[] getNext() {
            if ( numLeft.equals(total) ) {
                numLeft = numLeft.subtract (BigInteger.ONE);
                return a;
            }

            
            int j = a.length - 2;
            while ( a[j] > a[j+1] ) {
                j--;
            }

            
            int k = a.length - 1;
            while ( a[j] > a[k] ) {
                k--;
            }

            
            int temp = a[k];
            a[k] = a[j];
            a[j] = temp;

            
            int r = a.length - 1;
            int s = j + 1;
            while (r > s) {
                int temp2 = a[s];
                a[s] = a[r];
                a[r] = temp2;
                r--;
                s++;
            }

            numLeft = numLeft.subtract(BigInteger.ONE);
            return a;
        }
    }

}