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import java.math.BigInteger;
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import java.util.ArrayList;
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import java.util.Arrays;
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import java.util.List;
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public class LatinSquaresInReducedForm {
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public static void main(String[] args) {
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System.out.printf("Reduced latin squares of order 4:%n");
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for ( LatinSquare square : getReducedLatinSquares(4) ) {
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System.out.printf("%s%n", square);
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}
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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");
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for ( int n = 1 ; n <= 6 ; n++ ) {
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List<LatinSquare> list = getReducedLatinSquares(n);
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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));
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}
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}
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private static long fact(int n) {
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if ( n == 0 ) {
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return 1;
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}
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int prod = 1;
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for ( int i = 1 ; i <= n ; i++ ) {
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prod *= i;
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}
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return prod;
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}
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private static List<LatinSquare> getReducedLatinSquares(int n) {
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List<LatinSquare> squares = new ArrayList<>();
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squares.add(new LatinSquare(n));
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PermutationGenerator permGen = new PermutationGenerator(n);
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for ( int fillRow = 1 ; fillRow < n ; fillRow++ ) {
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List<LatinSquare> squaresNext = new ArrayList<>();
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for ( LatinSquare square : squares ) {
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while ( permGen.hasMore() ) {
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int[] perm = permGen.getNext();
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// If not the correct row - next permutation.
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if ( (perm[0]+1) != (fillRow+1) ) {
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continue;
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}
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// Check permutation against current square.
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boolean permOk = true;
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done:
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for ( int row = 0 ; row < fillRow ; row++ ) {
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for ( int col = 0 ; col < n ; col++ ) {
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if ( square.get(row, col) == (perm[col]+1) ) {
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permOk = false;
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break done;
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}
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}
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}
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if ( permOk ) {
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LatinSquare newSquare = new LatinSquare(square);
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for ( int col = 0 ; col < n ; col++ ) {
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newSquare.set(fillRow, col, perm[col]+1);
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}
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squaresNext.add(newSquare);
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}
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}
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permGen.reset();
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}
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squares = squaresNext;
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}
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return squares;
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}
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@SuppressWarnings("unused")
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private static int[] display(int[] in) {
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int [] out = new int[in.length];
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for ( int i = 0 ; i < in.length ; i++ ) {
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out[i] = in[i] + 1;
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}
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return out;
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}
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private static class LatinSquare {
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int[][] square;
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int size;
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public LatinSquare(int n) {
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square = new int[n][n];
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size = n;
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for ( int col = 0 ; col < n ; col++ ) {
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set(0, col, col + 1);
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}
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}
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public LatinSquare(LatinSquare ls) {
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int n = ls.size;
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square = new int[n][n];
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size = n;
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for ( int row = 0 ; row < n ; row++ ) {
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for ( int col = 0 ; col < n ; col++ ) {
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set(row, col, ls.get(row, col));
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}
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}
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}
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public void set(int row, int col, int value) {
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square[row][col] = value;
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}
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public int get(int row, int col) {
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return square[row][col];
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}
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@Override
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public String toString() {
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StringBuilder sb = new StringBuilder();
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for ( int row = 0 ; row < size ; row++ ) {
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sb.append(Arrays.toString(square[row]));
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sb.append("\n");
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}
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return sb.toString();
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}
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}
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private static class PermutationGenerator {
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private int[] a;
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private BigInteger numLeft;
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private BigInteger total;
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public PermutationGenerator (int n) {
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if (n < 1) {
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throw new IllegalArgumentException ("Min 1");
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}
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a = new int[n];
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total = getFactorial(n);
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reset();
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}
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private void reset () {
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for ( int i = 0 ; i < a.length ; i++ ) {
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a[i] = i;
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}
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numLeft = new BigInteger(total.toString());
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}
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public boolean hasMore() {
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return numLeft.compareTo(BigInteger.ZERO) == 1;
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}
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private static BigInteger getFactorial (int n) {
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BigInteger fact = BigInteger.ONE;
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for ( int i = n ; i > 1 ; i-- ) {
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fact = fact.multiply(new BigInteger(Integer.toString(i)));
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}
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return fact;
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}
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/*--------------------------------------------------------
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* Generate next permutation (algorithm from Rosen p. 284)
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*--------------------------------------------------------
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*/
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public int[] getNext() {
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if ( numLeft.equals(total) ) {
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numLeft = numLeft.subtract (BigInteger.ONE);
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return a;
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}
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// Find largest index j with a[j] < a[j+1]
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int j = a.length - 2;
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while ( a[j] > a[j+1] ) {
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j--;
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}
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// Find index k such that a[k] is smallest integer greater than a[j] to the right of a[j]
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int k = a.length - 1;
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while ( a[j] > a[k] ) {
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k--;
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}
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// Interchange a[j] and a[k]
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int temp = a[k];
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a[k] = a[j];
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a[j] = temp;
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// Put tail end of permutation after jth position in increasing order
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int r = a.length - 1;
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int s = j + 1;
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while (r > s) {
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int temp2 = a[s];
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a[s] = a[r];
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a[r] = temp2;
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r--;
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s++;
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}
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numLeft = numLeft.subtract(BigInteger.ONE);
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return a;
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}
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}
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}
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