Latin Squares in reduced form: Difference between revisions
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Demonstrate by:
* displaying the four reduced Latin Squares of order 4.
* for n = 1 to 6 (or more) produce the set of reduced Latin Squares; produce a table which shows the size of the set of reduced Latin Squares and compares this value times n! times (n-1)! with the values in [
<br><br>
=={{header|11l}}==
{{trans|Python}}
<syntaxhighlight lang="11l">F dList(n, =start)
start--
V a = Array(0 .< n)
a[start] = a[0]
a[0] = start
a.sort_range(1..)
V first = a[1]
[[Int]] r
F recurse(Int last) -> Void
I (last == @first)
L(v) @a[1..]
I L.index + 1 == v
R
V b = @a.map(x -> x + 1)
@r.append(b)
R
L(i) (last .< 0).step(-1)
swap(&@a[i], &@a[last])
@recurse(last - 1)
swap(&@a[i], &@a[last])
recurse(n - 1)
R r
F printSquare(latin, n)
L(row) latin
print(row)
print()
F reducedLatinSquares(n, echo)
I n <= 0
I echo
print(‘[]’)
R 0
E I n == 1
I echo
print([1])
R 1
V rlatin = [[0] * n] * n
L(j) 0 .< n
rlatin[0][j] = j + 1
V count = 0
F recurse(Int i) -> Void
V rows = dList(@n, i)
L(r) 0 .< rows.len
@rlatin[i - 1] = rows[r]
V justContinue = 0B
V k = 0
L !justContinue & k < i - 1
L(j) 1 .< @n
I @rlatin[k][j] == @rlatin[i - 1][j]
I r < rows.len - 1
justContinue = 1B
L.break
I i > 2
R
k++
I !justContinue
I i < @n
@recurse(i + 1)
E
@count++
I @echo
printSquare(@rlatin, @n)
recurse(2)
R count
print("The four reduced latin squares of order 4 are:\n")
reducedLatinSquares(4, 1B)
print(‘The size of the set of reduced latin squares for the following orders’)
print("and hence the total number of latin squares of these orders are:\n")
L(n) 1..6
V size = reducedLatinSquares(n, 0B)
V f = factorial(n - 1)
f *= f * n * size
print(‘Order #.: Size #<4 x #.! x #.! => Total #.’.format(n, size, n, n - 1, f))</syntaxhighlight>
{{out}}
<pre>
The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200
</pre>
=={{header|C sharp|C#}}==
{{trans|D}}
<syntaxhighlight lang="csharp">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--; // use 0 basing
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];
// recursive closure permutes a[1:]
matrix r = new matrix();
void recurse(int last) {
if (last == first) {
// bottom of recursion. you get here once for each permutation.
// test if permutation is deranged.
for (int j = 1; j < a.Length; j++) {
var v = a[j];
if (j == v) {
return; //no, ignore it
}
}
// yes, save a copy with 1 based indexing
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);
}
}
// first row
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: { }
}
}
//remaing rows
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);
}
}
}
}</syntaxhighlight>
{{out}}
<pre>The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|C++}}==
{{trans|C#}}
<syntaxhighlight lang="cpp">#include <algorithm>
#include <functional>
#include <iostream>
#include <numeric>
#include <vector>
typedef std::vector<std::vector<int>> matrix;
matrix dList(int n, int start) {
start--; // use 0 basing
std::vector<int> a(n);
std::iota(a.begin(), a.end(), 0);
a[start] = a[0];
a[0] = start;
std::sort(a.begin() + 1, a.end());
auto first = a[1];
// recursive closure permutes a[1:]
matrix r;
std::function<void(int)> recurse;
recurse = [&](int last) {
if (last == first) {
// bottom of recursion you get here once for each permutation.
// test if permutation is deranged.
for (size_t j = 1; j < a.size(); j++) {
auto v = a[j];
if (j == v) {
return; //no, ignore it
}
}
// yes, save a copy with 1 based indexing
std::vector<int> b;
std::transform(a.cbegin(), a.cend(), std::back_inserter(b), [](int v) { return v + 1; });
r.push_back(b);
return;
}
for (int i = last; i >= 1; i--) {
std::swap(a[i], a[last]);
recurse(last - 1);
std::swap(a[i], a[last]);
}
};
recurse(n - 1);
return r;
}
void printSquare(const matrix &latin, int n) {
for (auto &row : latin) {
auto it = row.cbegin();
auto end = row.cend();
std::cout << '[';
if (it != end) {
std::cout << *it;
it = std::next(it);
}
while (it != end) {
std::cout << ", " << *it;
it = std::next(it);
}
std::cout << "]\n";
}
std::cout << '\n';
}
unsigned long reducedLatinSquares(int n, bool echo) {
if (n <= 0) {
if (echo) {
std::cout << "[]\n";
}
return 0;
} else if (n == 1) {
if (echo) {
std::cout << "[1]\n";
}
return 1;
}
matrix rlatin;
for (int i = 0; i < n; i++) {
rlatin.push_back({});
for (int j = 0; j < n; j++) {
rlatin[i].push_back(j);
}
}
// first row
for (int j = 0; j < n; j++) {
rlatin[0][j] = j + 1;
}
unsigned long count = 0;
std::function<void(int)> recurse;
recurse = [&](int i) {
auto rows = dList(n, i);
for (size_t r = 0; r < rows.size(); 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.size() - 1) {
goto outer;
}
if (i > 2) {
return;
}
}
}
}
if (i < n) {
recurse(i + 1);
} else {
count++;
if (echo) {
printSquare(rlatin, n);
}
}
outer: {}
}
};
//remaining rows
recurse(2);
return count;
}
unsigned long factorial(unsigned long n) {
if (n <= 0) return 1;
unsigned long prod = 1;
for (unsigned long i = 2; i <= n; i++) {
prod *= i;
}
return prod;
}
int main() {
std::cout << "The four reduced lating squares of order 4 are:\n";
reducedLatinSquares(4, true);
std::cout << "The size of the set of reduced latin squares for the following orders\n";
std::cout << "and hence the total number of latin squares of these orders are:\n\n";
for (int n = 1; n < 7; n++) {
auto size = reducedLatinSquares(n, false);
auto f = factorial(n - 1);
f *= f * n * size;
std::cout << "Order " << n << ": Size " << size << " x " << n << "! x " << (n - 1) << "! => Total " << f << '\n';
}
return 0;
}</syntaxhighlight>
{{out}}
<pre>The four reduced lating squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|D}}==
{{trans|Go}}
<
import std.array;
import std.range;
Line 137 ⟶ 626:
writefln("Order %d: Size %-4d x %d! x %d! => Total %d", n, size, n, n - 1, f);
}
}</
{{out}}
<pre>The four reduced latin squares of order 4 are:
Line 174 ⟶ 663:
===The Function===
This task uses [[Permutations/Derangements#F.23]]
<
// Generate Latin Squares in reduced form. Nigel Galloway: July 10th., 2019
let normLS α=
Line 181 ⟶ 670:
let rec normLS n g=seq{for i in fG n N.[g] do if g=α-2 then yield [|1..α|]::(List.rev (i::n)) else yield! normLS (i::n) (g+1)}
match α with 1->seq[[[|1|]]] |2-> seq[[[|1;2|];[|2;1|]]] |_->Seq.collect(fun n->normLS [n] 1) N.[0]
</syntaxhighlight>
===The Task===
<
normLS 4 |> Seq.iter(fun n->List.iter(printfn "%A") n;printfn "");;
</syntaxhighlight>
{{out}}
<pre>
Line 208 ⟶ 697:
[|4; 3; 2; 1|]
</pre>
<
let rec fact n g=if n<2 then g else fact (n-1) n*g
[1..6] |> List.iter(fun n->let nLS=normLS n|>Seq.length in printfn "order=%d number of Reduced Latin Squares nLS=%d nLS*n!*(n-1)!=%d" n nLS (nLS*(fact n 1)*(fact (n-1) 1)))
</syntaxhighlight>
{{out}}
<pre>
Line 224 ⟶ 713:
=={{header|Go}}==
This reuses the dList function from the [[Permutations/Derangements#Go]] task, suitably adjusted for the present one.
<
import (
Line 360 ⟶ 849:
fmt.Printf("Order %d: Size %-4d x %d! x %d! => Total %d\n", n, size, n, n-1, f)
}
}</
{{out}}
Line 395 ⟶ 884:
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200
</pre>
=={{header|Haskell}}==
The solution uses permutation generator given by '''Data.List''' package and List monad for generating all possible latin squares as a fold of permutation list.
<syntaxhighlight lang="haskell">import Data.List (permutations, (\\))
import Control.Monad (foldM, forM_)
latinSquares :: Eq a => [a] -> [[[a]]]
latinSquares [] = []
latinSquares set = map reverse <$> squares
where
squares = foldM addRow firstRow perm
perm = tail (groupedPermutations set)
firstRow = pure <$> set
addRow tbl rows = [ zipWith (:) row tbl
| row <- rows
, and $ different (tail row) (tail tbl) ]
different = zipWith $ (not .) . elem
groupedPermutations :: Eq a => [a] -> [[[a]]]
groupedPermutations lst = map (\x -> (x :) <$> permutations (lst \\ [x])) lst
printTable :: Show a => [[a]] -> IO ()
printTable tbl = putStrLn $ unlines $ unwords . map show <$> tbl
</syntaxhighlight>
It is slightly optimized by grouping permutations by the first element according to a set order. Partitioning reduces the filtering procedure by factor of an initial set size.
'''Examples'''
<pre>λ> latinSquares "abc"
[["abc","bca","cab"]]
λ> mapM_ printTable $ take 3 $ latinSquares [1..9]
1 2 3 4 5 6 7 8 9
2 9 4 8 1 7 3 6 5
3 8 2 5 9 1 4 7 6
4 7 5 6 2 9 8 1 3
5 6 9 1 3 8 2 4 7
6 5 1 7 4 2 9 3 8
7 4 6 3 8 5 1 9 2
8 3 7 9 6 4 5 2 1
9 1 8 2 7 3 6 5 4
1 2 3 4 5 6 7 8 9
2 9 4 8 1 7 3 5 6
3 8 2 5 9 1 4 6 7
4 7 5 6 2 9 8 1 3
5 6 9 1 3 8 2 7 4
6 5 1 7 4 2 9 3 8
7 4 6 3 8 5 1 9 2
8 3 7 9 6 4 5 2 1
9 1 8 2 7 3 6 4 5
1 2 3 4 5 6 7 8 9
2 9 4 8 1 7 3 6 5
3 8 2 5 9 1 4 7 6
4 7 5 6 2 9 1 3 8
5 6 9 1 3 8 2 4 7
6 5 1 7 4 2 8 9 3
7 4 6 3 8 5 9 1 2
8 3 7 9 6 4 5 2 1
9 1 8 2 7 3 6 5 4</pre>
'''Tasks'''
<syntaxhighlight lang="haskell">task1 = do
putStrLn "Latin squares of order 4:"
mapM_ printTable $ latinSquares [1..4]
task2 = do
putStrLn "Sizes of latin squares sets for different orders:"
forM_ [1..6] $ \n ->
let size = length $ latinSquares [1..n]
total = fact n * fact (n-1) * size
fact i = product [1..i]
in printf "Order %v: %v*%v!*%v!=%v\n" n size n (n-1) total</syntaxhighlight>
<pre>λ> task1 >> task2
Latin squares of order 4:
1 2 3 4
4 1 2 3
3 4 1 2
2 3 4 1
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
1 2 3 4
2 1 4 3
4 3 1 2
3 4 2 1
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
Sizes of latin squares sets for different orders:
Order 1: 1*1!*0!=1
Order 2: 1*2!*1!=2
Order 3: 1*3!*2!=12
Order 4: 4*4!*3!=576
Order 5: 56*5!*4!=161280
Order 6: 9408*6!*5!=812851200</pre>
=={{header|J}}==
Implementation:
<syntaxhighlight lang="j">
redlat=: {{
perms=: (A.&i.~ !)~ y
sqs=. i.1 1,y
for_j.}.i.y do.
p=. (j={."1 perms)#perms
sel=.-.+./"1 p +./@:="1/"2 sqs
sqs=.(#~ 1-0*/ .="1{:"2),/sqs,"2 1 sel#"2 p
end.
}}
</syntaxhighlight>
Task examples:
<syntaxhighlight lang="j"> redlat 4
0 1 2 3
1 0 3 2
2 3 0 1
3 2 1 0
0 1 2 3
1 0 3 2
2 3 1 0
3 2 0 1
0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2
0 1 2 3
1 3 0 2
2 0 3 1
3 2 1 0
#@redlat every 1 2 3 4 5 6
1 1 1 4 56 9408
(#@redlat every 1 2 3 4 5 6)*(!1 2 3 4 5 6x)*(!0 1 2 3 4 5x)
1 2 12 576 161280 812851200
</syntaxhighlight>
=={{header|Java}}==
<syntaxhighlight lang="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 not the correct row - next permutation.
if ( (perm[0]+1) != (fillRow+1) ) {
continue;
}
// Check permutation against current square.
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;
}
/*--------------------------------------------------------
* Generate next permutation (algorithm from Rosen p. 284)
*--------------------------------------------------------
*/
public int[] getNext() {
if ( numLeft.equals(total) ) {
numLeft = numLeft.subtract (BigInteger.ONE);
return a;
}
// Find largest index j with a[j] < a[j+1]
int j = a.length - 2;
while ( a[j] > a[j+1] ) {
j--;
}
// Find index k such that a[k] is smallest integer greater than a[j] to the right of a[j]
int k = a.length - 1;
while ( a[j] > a[k] ) {
k--;
}
// Interchange a[j] and a[k]
int temp = a[k];
a[k] = a[j];
a[j] = temp;
// Put tail end of permutation after jth position in increasing order
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;
}
}
}
</syntaxhighlight>
{{out}}
<pre>
Reduced latin squares of order 4:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
Compute the number of latin squares from count of reduced latin squares:
(Reduced Latin Square Count) * n! * (n-1)! = Latin Square Count
Size = 1, 1 * 1 * 1 = 1
Size = 2, 1 * 2 * 1 = 2
Size = 3, 1 * 6 * 2 = 12
Size = 4, 4 * 24 * 6 = 576
Size = 5, 56 * 120 * 24 = 161,280
Size = 6, 9408 * 720 * 120 = 812,851,200
</pre>
=={{header|jq}}==
{{works with|jq}}
'''Works with recent versions of gojq''' (e.g. f0faa22 (August 22, 2021))
''' Preliminaries'''
<syntaxhighlight lang="jq">def count(s): reduce s as $x (0; .+1);
def factorial: reduce range(2;.+1) as $i (1; . * $i);
def permutations:
if length == 0 then []
else
range(0;length) as $i
| [.[$i]] + (del(.[$i])|permutations)
end ;
</syntaxhighlight>
'''Latin Squares'''
<syntaxhighlight lang="jq">def clash($row2; $row1):
any(range(0;$row2|length); $row1[.] == $row2[.]);
# Input is a row; stream is a stream of rows
def clash(stream):
. as $row | any(stream; clash($row; .)) ;
# Emit a stream of latin squares of size .
def latin_squares:
. as $n
# Emit a stream of arrays of permutation of 1 .. $n inclusive, and beginning with $i
| def permutations_beginning_with($i):
[$i] + ([range(1; $i), range($i+1; $n + 1)] | permutations);
# input: an array of rows, $rows
# output: a stream of all the permutations starting with $i
# that are permissible relative to $rows
def filter_permuted($i):
. as $rows
| permutations_beginning_with($i)
| select( clash($rows[]) | not ) ;
# input: an array of the first few rows (at least one) of a latin square
# output: a stream of possible immediate-successor rows
def next_latin_square_row:
filter_permuted(1 + .[-1][0]);
# recursion makes completing a latin square a snap
def complete_latin_square:
if length == $n then .
else next_latin_square_row as $next
| . + [$next] | complete_latin_square
end;
[[range(1;$n+1)]]
| complete_latin_square ;
</syntaxhighlight>
'''The Task'''
<syntaxhighlight lang="jq">def task:
"The reduced latin squares of order 4 are:",
(4 | latin_squares),
"",
(range(1; 7)
| . as $i
| count(latin_squares) as $c
| ($c * factorial * ((.-1)|factorial)) as $total
| "There are \($c) reduced latin squares of order \(.); \($c) * \(.)! * \(.-1)! is \($total)"
) ;
task</syntaxhighlight>
{{out}}
Invocation: jq -nrc -f latin-squares.jq
<pre>
The reduced latin squares of order 4 are:
[[1,2,3,4],[2,1,4,3],[3,4,1,2],[4,3,2,1]]
[[1,2,3,4],[2,1,4,3],[3,4,2,1],[4,3,1,2]]
[[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]]
[[1,2,3,4],[2,4,1,3],[3,1,4,2],[4,3,2,1]]
There are 1 reduced latin squares of order 1; 1 * 1! * 0! is 1
There are 1 reduced latin squares of order 2; 1 * 2! * 1! is 2
There are 1 reduced latin squares of order 3; 1 * 3! * 2! is 12
There are 4 reduced latin squares of order 4; 4 * 4! * 3! is 576
There are 56 reduced latin squares of order 5; 56 * 5! * 4! is 161280
There are 9408 reduced latin squares of order 6; 9408 * 6! * 5! is 812851200
</pre>
=={{header|Julia}}==
<syntaxhighlight lang="julia">using Combinatorics
clash(row2, row1::Vector{Int}) = any(i -> row1[i] == row2[i], 1:length(row2))
clash(row, rows::Vector{Vector{Int}}) = any(r -> clash(row, r), rows)
permute_onefixed(i, n) = map(vec -> vcat(i, vec), permutations(filter(x -> x != i, 1:n)))
filter_permuted(rows, i, n) = filter(v -> !clash(v, rows), permute_onefixed(i, n))
function makereducedlatinsquares(n)
matarray = [reshape(collect(1:n), 1, n)]
for i in 2:n
newmatarray = Vector{Matrix{Int}}()
for mat in matarray
r = size(mat)[1] + 1
newrows = filter_permuted(collect(row[:] for row in eachrow(mat)), r, n)
newmat = zeros(Int, r, n)
newmat[1:r-1, :] .= mat
append!(newmatarray,
[deepcopy(begin newmat[i, :] .= row; newmat end) for row in newrows])
end
matarray = newmatarray
end
matarray, length(matarray)
end
function testlatinsquares()
squares, count = makereducedlatinsquares(4)
println("The four reduced latin squares of order 4 are:")
for sq in squares, (i, row) in enumerate(eachrow(sq)), j in 1:4
print(row[j], j == 4 ? (i == 4 ? "\n\n" : "\n") : " ")
end
for i in 1:6
squares, count = makereducedlatinsquares(i)
println("Order $i: Size ", rpad(count, 5), "* $(i)! * $(i - 1)! = ",
count * factorial(i) * factorial(i - 1))
end
end
testlatinsquares()
</syntaxhighlight>{{out}}
<pre>
The four reduced latin squares of order 4 are:
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
1 2 3 4
2 1 4 3
3 4 2 1
4 3 1 2
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
Order 1: Size 1 * 1! * 0! = 1
Order 2: Size 1 * 2! * 1! = 2
Order 3: Size 1 * 3! * 2! = 12
Order 4: Size 4 * 4! * 3! = 576
Order 5: Size 56 * 5! * 4! = 161280
Order 6: Size 9408 * 6! * 5! = 812851200
</pre>
=={{header|Kotlin}}==
{{trans|D}}
<
fun dList(n: Int, sp: Int): Matrix {
Line 519 ⟶ 1,557:
println("Order $n: Size %-4d x $n! x ${n - 1}! => Total $f".format(size))
}
}</
{{out}}
<pre>The four reduced latin squares of order 4 are:
Line 552 ⟶ 1,590:
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|MiniZinc}}==
===The Model (lsRF.mnz)===
<syntaxhighlight lang="minizinc">
%Latin Squares in Reduced Form. Nigel Galloway, September 5th., 2019
include "alldifferent.mzn";
int: N;
array[1..N,1..N] of var 1..N: p; constraint forall(n in 1..N)(p[1,n]=n /\ p[n,1]=n);
constraint forall(n in 1..N)(alldifferent([p[n,g]|g in 1..N])/\alldifferent([p[g,n]|g in 1..N]));
</syntaxhighlight>
===The Tasks===
;displaying the four reduced Latin Squares of order 4
<syntaxhighlight lang="minizinc">
include "lsRF.mzn";
output [show_int(1,p[i,j])++
if j == 4 then
if i != 4 then "\n"
else "" endif
else "" endif
| i,j in 1..4 ] ++ ["\n"];
</syntaxhighlight>
When the above is run using minizinc --all-solutions -DN=4 the following is produced:
{{out}}
<pre>
1234
2143
3421
4312
----------
1234
2143
3412
4321
----------
1234
2413
3142
4321
----------
1234
2341
3412
4123
----------
==========
</pre>
;counting the solutions
minizinc.exe --all-solutions -DN=5 -s lsRF.mzn produces the following:
<pre>
.
.
.
p = array2d(1..5, 1..5, [1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 3, 1, 5, 2, 4, 4, 5, 2, 1, 3, 5, 4, 1, 3, 2]);
----------
p = array2d(1..5, 1..5, [1, 2, 3, 4, 5, 2, 3, 5, 1, 4, 3, 5, 4, 2, 1, 4, 1, 2, 5, 3, 5, 4, 1, 3, 2]);
----------
p = array2d(1..5, 1..5, [1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 3, 5, 2, 1, 4, 4, 1, 5, 2, 3, 5, 4, 1, 3, 2]);
----------
==========
%%%mzn-stat: initTime=0.057
%%%mzn-stat: solveTime=0.003
%%%mzn-stat: solutions=56
%%%mzn-stat: variables=43
%%%mzn-stat: propagators=8
%%%mzn-stat: propagations=960
%%%mzn-stat: nodes=111
%%%mzn-stat: failures=0
%%%mzn-stat: restarts=0
%%%mzn-stat: peakDepth=7
%%%mzn-stat-end
%%%mzn-stat: nSolutions=56
</pre>
and minizinc.exe --all-solutions -DN=6 -s lsRF.mzn produces the following:
<pre>
.
.
.
p = array2d(1..6, 1..6, [1, 2, 3, 4, 5, 6, 2, 4, 5, 6, 3, 1, 3, 1, 4, 2, 6, 5, 4, 6, 2, 5, 1, 3, 5, 3, 6, 1, 2, 4, 6, 5, 1, 3, 4, 2]);
----------
p = array2d(1..6, 1..6, [1, 2, 3, 4, 5, 6, 2, 1, 4, 6, 3, 5, 3, 4, 5, 2, 6, 1, 4, 6, 2, 5, 1, 3, 5, 3, 6, 1, 2, 4, 6, 5, 1, 3, 4, 2]);
----------
==========
%%%mzn-stat: initTime=0.003
%%%mzn-stat: solveTime=6.669
%%%mzn-stat: solutions=9408
%%%mzn-stat: variables=58
%%%mzn-stat: propagators=10
%%%mzn-stat: propagations=179635
%%%mzn-stat: nodes=19035
%%%mzn-stat: failures=110
%%%mzn-stat: restarts=0
%%%mzn-stat: peakDepth=17
%%%mzn-stat-end
%%%mzn-stat: nSolutions=9408
</pre>
The only way to complete the tasks requirement to produce a table is with another language. Ruby has the ability to run an external program, capture the output, and text handling ability to format it to this tasks requirements. Othe scripting languages are available.
=={{header|Nim}}==
{{trans|Go, Python, D, Kotlin}}
We use the Go algorithm but have chosen to create two types, Row and Matrix, to simulate sequences starting at index 1. So, the indexes and tests are somewhat different.
<syntaxhighlight lang="nim">import algorithm, math, sequtils, strformat
type
# Row managed as a sequence of ints with base index 1.
Row = object
value: seq[int]
# Matrix managed as a sequence of rows with base index 1.
Matrix = object
value: seq[Row]
func newRow(n: Natural = 0): Row =
## Create a new row of length "n".
Row(value: newSeq[int](n))
# Create a new matrix of length "n" containing rows of length "p".
func newMatrix(n, p: Natural = 0): Matrix = Matrix(value: newSeqWith(n, newRow(p)))
# Functions for rows.
func `[]`(r: var Row; i: int): var int = r.value[i - 1]
func `[]=`(r: var Row; i, n: int) = r.value[i - 1] = n
func sort(r: var Row; low, high: Positive) =
r.value.toOpenArray(low - 1, high - 1).sort()
func `$`(r: Row): string = ($r.value)[1..^1]
# Functions for matrices.
func `[]`(m: Matrix; i: int): Row = m.value[i - 1]
func `[]`(m: var Matrix; i: int): var Row = m.value[i - 1]
func `[]=`(m: var Matrix; i: int; r: Row) = m.value[i - 1] = r
func high(m: Matrix): Natural = m.value.len
func add(m: var Matrix; r: Row) = m.value.add r
func `$`(m: Matrix): string =
for row in m.value: result.add $row & '\n'
func dList(n, start: Positive): Matrix =
## Generate derangements of first 'n' numbers, with 'start' in first place.
var a = Row(value: toSeq(1..n))
swap a[1], a[start]
a.sort(2, n)
let first = a[2]
var r: Matrix
func recurse(last: int) =
## Recursive closure permutes a[2..^1].
if last == first:
# Bottom of recursion. You get here once for each permutation.
# Test if permutation is deranged.
for i in 2..n:
if a[i] == i: return # No: ignore it.
r.add a
return
for i in countdown(last, 2):
swap a[i], a[last]
recurse(last - 1)
swap a[i], a[last]
recurse(n)
result = r
proc reducedLatinSquares(n: Positive; print: bool): int =
if n == 1:
if print: echo [1]
return 1
var rlatin = newMatrix(n, n)
# Initialize first row.
for i in 1..n: rlatin[1][i] = i
var count = 0
proc recurse(i: int) =
let rows = dList(n, i)
for r in 1..rows.high:
block inner:
rlatin[i] = rows[r]
for k in 1..<i:
for j in 2..n:
if rlatin[k][j] == rlatin[i][j]:
if r < rows.high: break inner
if i > 2: return
if i < n:
recurse(i + 1)
else:
inc count
if print: echo rlatin
# Remaining rows.
recurse(2)
result = count
when isMainModule:
echo "The four reduced latin squares of order 4 are:"
discard reducedLatinSquares(4, true)
echo "The size of the set of reduced latin squares for the following orders"
echo "and hence the total number of latin squares of these orders are:"
for n in 1..6:
let size = reducedLatinSquares(n, false)
let f = fac(n - 1)^2 * n * size
echo &"Order {n}: Size {size:<4} x {n}! x {n - 1}! => Total {f}"</syntaxhighlight>
{{out}}
<pre>The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|Perl}}==
It takes a little under 2 minutes to find order 7.
<syntaxhighlight lang="perl">#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Latin_Squares_in_reduced_form
use warnings;
my $n = 0;
my $count;
our @perms;
while( ++$n <= 7 )
{
$count = 0;
@perms = perm( my $start = join '', 1 .. $n );
find( $start );
print "order $n size $count total @{[$count * fact($n) * fact($n-1)]}\n\n";
}
sub find
{
@_ >= $n and return $count += ($n != 4) || print join "\n", @_, "\n";
local @perms = grep 0 == ($_[-1] ^ $_) =~ tr/\0//, @perms;
my $row = @_ + 1;
find( @_, $_ ) for grep /^$row/, @perms;
}
sub fact { $_[0] > 1 ? $_[0] * fact($_[0] - 1) : 1 }
sub perm
{
my $s = shift;
length $s <= 1 ? $s :
map { my $f = $_; map "$f$_", perm( $s =~ s/$_//r ) } split //, $s;
}</syntaxhighlight>
{{out}}
<pre>
order 1 size 1 total 1
order 2 size 1 total 2
order 3 size 1 total 12
1234
2143
3412
4321
1234
2143
3421
4312
1234
2341
3412
4123
1234
2413
3142
4321
order 4 size 4 total 576
order 5 size 56 total 161280
order 6 size 9408 total 812851200
order 7 size 16942080 total 61479419904000
</pre>
=={{header|Phix}}==
A Simple backtracking search.<br>
aside: in phix here is no difference between res[r][c] and res[r,c]. I mixed them here, using whichever felt the more natural to me.
<!--<syntaxhighlight lang="phix">-->
<span style="color: #004080;">string</span> <span style="color: #000000;">aleph</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">rfls</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">bool</span> <span style="color: #000000;">count_only</span><span style="color: #0000FF;">=</span><span style="color: #004600;">true</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">aleph</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- too big...</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">count_only</span><span style="color: #0000FF;">?</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:{{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}})</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">tn</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span> <span style="color: #000080;font-style:italic;">-- {1..n}</span>
<span style="color: #000000;">vcs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span> <span style="color: #000080;font-style:italic;">-- valid for cols</span>
<span style="color: #000000;">vrs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span> <span style="color: #000080;font-style:italic;">-- valid for rows</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (main workspace/one element of result)</span>
<span style="color: #004080;">object</span> <span style="color: #000000;">result</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">count_only</span><span style="color: #0000FF;">?</span><span style="color: #000000;">0</span><span style="color: #0000FF;">:{})</span>
<span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> <span style="color: #000080;font-style:italic;">-- (not strictly necessary)</span>
<span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> <span style="color: #000080;font-style:italic;">-- """</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span> <span style="color: #0000FF;">&</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<span style="color: #000080;font-style:italic;">-- place with backtrack:
-- if we successfully place [n,n] add to results and backtrack
-- terminate when we fail to place or backtrack from [2,2]</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">rrc</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">rrc</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- backtrack (/undo)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rrc</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- sanity check</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">c</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rrc</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- ""</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rrc</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">rrc</span>
<span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">c</span><span style="color: #0000FF;">][</span><span style="color: #000000;">rrc</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">rrc</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">found</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">rrc</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">and</span> <span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">c</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span>
<span style="color: #000000;">vrs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #000000;">vcs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">c</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #000000;">found</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #008080;">exit</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">found</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">n</span> <span style="color: #008080;">and</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">=</span><span style="color: #000000;">n</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">count_only</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">result</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">else</span>
<span style="color: #000000;">result</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">result</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000080;font-style:italic;">-- (here, backtracking == not advancing)</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">=</span><span style="color: #000000;">n</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">else</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">else</span>
<span style="color: #000080;font-style:italic;">-- backtrack</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">and</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">result</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">reduced_form_latin_squares</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">rfls</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #004600;">false</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">line</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">line</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">aleph</span><span style="color: #0000FF;">[</span><span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">line</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"There are %d reduced form latin squares of order %d:\n%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">),</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000000;">reduced_form_latin_squares</span><span style="color: #0000FF;">(</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">6</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">size</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">rfls</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">size</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Order %d: Size %-4d x %d! x %d! => Total %d\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">size</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Line 675 ⟶ 2,027:
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200
</pre>
Whle the above finishes near-instantly, if you push it to 7 and add an elapsed(), you'll get:
<pre>
Order 7: Size 16942080 x 7! x 6! => Total 61479419904000
"2 minutes and 23s"
</pre>
=={{header|Picat}}==
Using Constraint modelling.
====The four solutions for N=4====
<syntaxhighlight lang="picat">import cp.
main =>
N = 4,
latin_square_reduced_form(N, X),
foreach(Row in X)
println(Row.to_list)
end,
nl,
fail.
latin_square_reduced_form(N, X) =>
X = new_array(N,N),
X :: 1..N,
foreach(I in 1..N)
all_different([X[I,J] : J in 1..N]),
all_different([X[J,I] : J in 1..N]),
X[1,I] #= I,
X[I,1] #= I
end,
solve(X).</syntaxhighlight>
{{out}}
<pre>[1,2,3,4]
[2,1,4,3]
[3,4,1,2]
[4,3,2,1]
[1,2,3,4]
[2,1,4,3]
[3,4,2,1]
[4,3,1,2]
[1,2,3,4]
[2,3,4,1]
[3,4,1,2]
[4,1,2,3]
[1,2,3,4]
[2,4,1,3]
[3,1,4,2]
[4,3,2,1]</pre>
====Number of solutions====
<syntaxhighlight lang="picat">import cp.
main =>
foreach(N in 1..7)
Count = count_all(latin_square_reduced_form(N, _X)),
printf("%2d %10d x %d! x %d! %16w\n",N,Count,N,N-1, Count*factorial(N)*factorial(N-1))
end,
nl.</syntaxhighlight>
{{out}}
<pre> 1 1 x 1! x 0! 1
2 1 x 2! x 1! 2
3 1 x 3! x 2! 12
4 4 x 4! x 3! 576
5 56 x 5! x 4! 161280
6 9408 x 6! x 5! 812851200
7 16942080 x 7! x 6! 61479419904000</pre>
For N=1..6 this model takes 23ms. For N=1..7 it takes 28.1s.
=={{header|Python}}==
{{trans|D}}
<syntaxhighlight lang="python">def dList(n, start):
start -= 1 # use 0 basing
a = range(n)
a[start] = a[0]
a[0] = start
a[1:] = sorted(a[1:])
first = a[1]
# rescursive closure permutes a[1:]
r = []
def recurse(last):
if (last == first):
# bottom of recursion. you get here once for each permutation.
# test if permutation is deranged.
# yes, save a copy with 1 based indexing
for j,v in enumerate(a[1:]):
if j + 1 == v:
return # no, ignore it
b = [x + 1 for x in a]
r.append(b)
return
for i in xrange(last, 0, -1):
a[i], a[last] = a[last], a[i]
recurse(last - 1)
a[i], a[last] = a[last], a[i]
recurse(n - 1)
return r
def printSquare(latin,n):
for row in latin:
print row
print
def reducedLatinSquares(n,echo):
if n <= 0:
if echo:
print []
return 0
elif n == 1:
if echo:
print [1]
return 1
rlatin = [None] * n
for i in xrange(n):
rlatin[i] = [None] * n
# first row
for j in xrange(0, n):
rlatin[0][j] = j + 1
class OuterScope:
count = 0
def recurse(i):
rows = dList(n, i)
for r in xrange(len(rows)):
rlatin[i - 1] = rows[r]
justContinue = False
k = 0
while not justContinue and k < i - 1:
for j in xrange(1, n):
if rlatin[k][j] == rlatin[i - 1][j]:
if r < len(rows) - 1:
justContinue = True
break
if i > 2:
return
k += 1
if not justContinue:
if i < n:
recurse(i + 1)
else:
OuterScope.count += 1
if echo:
printSquare(rlatin, n)
# remaining rows
recurse(2)
return OuterScope.count
def factorial(n):
if n == 0:
return 1
prod = 1
for i in xrange(2, n + 1):
prod *= i
return prod
print "The four reduced latin squares of order 4 are:\n"
reducedLatinSquares(4,True)
print "The size of the set of reduced latin squares for the following orders"
print "and hence the total number of latin squares of these orders are:\n"
for n in xrange(1, 7):
size = reducedLatinSquares(n, False)
f = factorial(n - 1)
f *= f * n * size
print "Order %d: Size %-4d x %d! x %d! => Total %d" % (n, size, n, n - 1, f)</syntaxhighlight>
{{out}}
<pre>The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|Raku}}==
(formerly Perl 6)
<syntaxhighlight lang="raku" line># utilities: factorial, sub-factorial, derangements
sub postfix:<!>($n) { (constant f = 1, |[\×] 1..*)[$n] }
sub prefix:<!>($n) { (1, 0, 1, -> $a, $b { ($++ + 2) × ($b + $a) } ... *)[$n] }
sub derangements(@l) { @l.permutations.grep(-> @p { none(@p Zeqv @l) }) }
sub LS-reduced (Int $n) {
return [1] if $n == 1;
my @LS;
my @l = 1 X+ ^$n;
my %D = derangements(@l).classify(*.[0]);
for [X] (^(!$n/($n-1))) xx $n-1 -> $tuple {
my @d.push: @l;
@d.push: %D{2}[$tuple[0]];
LOOP:
for 3 .. $n -> $x {
my @try = |%D{$x}[$tuple[$x-2]];
last LOOP if any @try »==« @d[$_] for 1..@d-1;
@d.push: @try;
}
next unless @d == $n and [==] [Z+] @d;
@LS.push: @d;
}
@LS
}
say .join("\n") ~ "\n" for LS-reduced(4);
for 1..6 -> $n {
printf "Order $n: Size %-4d x $n! x {$n-1}! => Total %d\n", $_, $_ * $n! * ($n-1)! given LS-reduced($n).elems
}</syntaxhighlight>
{{out}}
<pre>1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
1 2 3 4
2 1 4 3
3 4 2 1
4 3 1 2
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|Ruby}}==
{{trans|D}}
<syntaxhighlight lang="ruby">def printSquare(a)
for row in a
print row, "\n"
end
print "\n"
end
def dList(n, start)
start = start - 1 # use 0 based indexing
a = Array.new(n) {|i| i}
a[0], a[start] = a[start], a[0]
a[1..] = a[1..].sort
first = a[1]
r = []
recurse = lambda {|last|
if last == first then
# bottom of recursion, reached once for each permutation
# test if permutation is deranged
a[1..].each_with_index {|v, j|
if j + 1 == v then
return # no, ignore it
end
}
# yes, save a copy with 1 based indexing
b = a.map { |i| i + 1 }
r << b
return
end
i = last
while i >= 1 do
a[i], a[last] = a[last], a[i]
recurse.call(last - 1)
a[i], a[last] = a[last], a[i]
i = i - 1
end
}
recurse.call(n - 1)
return r
end
def reducedLatinSquares(n, echo)
if n <= 0 then
if echo then
print "[]\n\n"
end
return 0
end
if n == 1 then
if echo then
print "[1]\n\n"
end
return 1
end
rlatin = Array.new(n) { Array.new(n, Float::NAN)}
# first row
for j in 0 .. n - 1
rlatin[0][j] = j + 1
end
count = 0
recurse = lambda {|i|
rows = dList(n, i)
for r in 0 .. rows.length - 1
rlatin[i - 1] = rows[r].dup
catch (:outer) do
for k in 0 .. i - 2
for j in 1 .. n - 1
if rlatin[k][j] == rlatin[i - 1][j] then
if r < rows.length - 1 then
throw :outer
end
if i > 2 then
return
end
end
end
end
if i < n then
recurse.call(i + 1)
else
count = count + 1
if echo then
printSquare(rlatin)
end
end
end
end
}
# remaining rows
recurse.call(2)
return count
end
def factorial(n)
if n == 0 then
return 1
end
prod = 1
for i in 2 .. n
prod = prod * i
end
return prod
end
print "The four reduced latin squares of order 4 are:\n"
reducedLatinSquares(4, true)
print "The size of the set of reduced latin squares for the following orders\n"
print "and hence the total number of latin squares of these orders are:\n"
for n in 1 .. 6
size = reducedLatinSquares(n, false)
f = factorial(n - 1)
f = f * f * n * size
print "Order %d Size %-4d x %d! x %d! => Total %d\n" % [n, size, n, n - 1, f]
end</syntaxhighlight>
{{out}}
<pre>The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1 Size 1 x 1! x 0! => Total 1
Order 2 Size 1 x 2! x 1! => Total 2
Order 3 Size 1 x 3! x 2! => Total 12
Order 4 Size 4 x 4! x 3! => Total 576
Order 5 Size 56 x 5! x 4! => Total 161280
Order 6 Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|Visual Basic .NET}}==
{{trans|C#}}
<syntaxhighlight lang="vbnet">Option Strict On
Imports Matrix = System.Collections.Generic.List(Of System.Collections.Generic.List(Of Integer))
Module Module1
Sub Swap(Of T)(ByRef a As T, ByRef b As T)
Dim u = a
a = b
b = u
End Sub
Sub PrintSquare(latin As Matrix)
For Each row In latin
Dim it = row.GetEnumerator
Console.Write("[")
If it.MoveNext Then
Console.Write(it.Current)
End If
While it.MoveNext
Console.Write(", ")
Console.Write(it.Current)
End While
Console.WriteLine("]")
Next
Console.WriteLine()
End Sub
Function DList(n As Integer, start As Integer) As Matrix
start -= 1 REM use 0 based indexes
Dim a = Enumerable.Range(0, n).ToArray
a(start) = a(0)
a(0) = start
Array.Sort(a, 1, a.Length - 1)
Dim first = a(1)
REM recursive closure permutes a[1:]
Dim r As New Matrix
Dim Recurse As Action(Of Integer) = Sub(last As Integer)
If last = first Then
REM bottom of recursion. you get here once for each permutation
REM test if permutation is deranged.
For j = 1 To a.Length - 1
Dim v = a(j)
If j = v Then
Return REM no, ignore it
End If
Next
REM yes, save a copy with 1 based indexing
Dim b = a.Select(Function(v) v + 1).ToArray
r.Add(b.ToList)
Return
End If
For i = last To 1 Step -1
Swap(a(i), a(last))
Recurse(last - 1)
Swap(a(i), a(last))
Next
End Sub
Recurse(n - 1)
Return r
End Function
Function ReducedLatinSquares(n As Integer, echo As Boolean) As ULong
If n <= 0 Then
If echo Then
Console.WriteLine("[]")
Console.WriteLine()
End If
Return 0
End If
If n = 1 Then
If echo Then
Console.WriteLine("[1]")
Console.WriteLine()
End If
Return 1
End If
Dim rlatin As New Matrix
For i = 0 To n - 1
rlatin.Add(New List(Of Integer))
For j = 0 To n - 1
rlatin(i).Add(0)
Next
Next
REM first row
For j = 0 To n - 1
rlatin(0)(j) = j + 1
Next
Dim count As ULong = 0
Dim Recurse As Action(Of Integer) = Sub(i As Integer)
Dim rows = DList(n, i)
For r = 0 To rows.Count - 1
rlatin(i - 1) = rows(r)
For k = 0 To i - 2
For j = 1 To n - 1
If rlatin(k)(j) = rlatin(i - 1)(j) Then
If r < rows.Count - 1 Then
GoTo outer
End If
If i > 2 Then
Return
End If
End If
Next
Next
If i < n Then
Recurse(i + 1)
Else
count += 1UL
If echo Then
PrintSquare(rlatin)
End If
End If
outer:
While False
REM empty
End While
Next
End Sub
REM remiain rows
Recurse(2)
Return count
End Function
Function Factorial(n As ULong) As ULong
If n <= 0 Then
Return 1
End If
Dim prod = 1UL
For i = 2UL To n
prod *= i
Next
Return prod
End Function
Sub Main()
Console.WriteLine("The four reduced latin squares of order 4 are:")
Console.WriteLine()
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:")
Console.WriteLine()
For n = 1 To 6
Dim nu As ULong = CULng(n)
Dim size = ReducedLatinSquares(n, False)
Dim f = Factorial(nu - 1UL)
f *= f * nu * size
Console.WriteLine("Order {0}: Size {1} x {2}! x {3}! => Total {4}", n, size, n, n - 1, f)
Next
End Sub
End Module</syntaxhighlight>
{{out}}
<pre>The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200</pre>
=={{header|Wren}}==
{{trans|Go}}
{{libheader|Wren-sort}}
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./sort" for Sort
import "./math" for Int
import "./fmt" for Fmt
// generate derangements of first n numbers, with 'start' in first place.
var dList = Fn.new { |n, start|
var r = []
start = start - 1 // use 0 basing
var a = [0] * n
for (i in 1...n) a[i] = i
a[start] = a[0]
a[0] = start
Sort.quick(a, 1, a.count - 1, false)
var first = a[1]
var recurse // recursive closure permutes a[1..-1]
recurse = Fn.new { |last|
if (last == first) {
// bottom of recursion. you get here once for each permutation.
// test if permutation is deranged.
var j = 1
for (v in a.skip(1)) {
if (j == v) return // no, ignore it
j = j + 1
}
// yes, save a copy
var b = a.toList
for (i in 0...b.count) b[i] = b[i] + 1 // change back to 1 basing
r.add(b)
return
}
var i = last
while (i >= 1) {
var t = a[i]
a[i] = a[last]
a[last] = t
recurse.call(last-1)
t = a[i]
a[i] = a[last]
a[last] = t
i = i - 1
}
}
recurse.call(n-1)
return r
}
var printSquare = Fn.new { |latin, n|
System.print(latin.join("\n"))
System.print()
}
var reducedLatinSquare = Fn.new { |n, echo|
if (n <= 0) {
if (echo) System.print("[]\n")
return 0
}
if (n == 1) {
if (echo) System.print("[1]\n")
return 1
}
var rlatin = List.filled(n, null)
for (i in 0...n) rlatin[i] = List.filled(n, 0)
// first row
for (j in 0...n) rlatin[0][j] = j + 1
var count = 0
var recurse // // recursive closure to compute reduced latin squares and count or print them
recurse = Fn.new { |i|
var rows = dList.call(n, i) // get derangements of first n numbers, with 'i' first.
for (r in 0...rows.count) {
var outer = false
for (rr in 0...rows[r].count) rlatin[i-1][rr] = rows[r][rr]
var k = 0
while (k < i-1) {
var j = 1
while (j < n) {
if (rlatin[k][j] == rlatin[i-1][j]) {
if (r < rows.count - 1) {
outer = true
break
} else if (i > 2) {
return
}
}
j = j + 1
}
if (outer) break
k = k + 1
}
if (!outer) {
if (i < n) {
recurse.call(i + 1)
} else {
count = count + 1
if (echo) printSquare.call(rlatin, n)
}
}
}
}
// remaining rows
recurse.call(2)
return count
}
System.print("The four reduced latin squares of order 4 are:\n")
reducedLatinSquare.call(4, true)
System.print("The size of the set of reduced latin squares for the following orders")
System.print("and hence the total number of latin squares of these orders are:\n")
for (n in 1..6) {
var size = reducedLatinSquare.call(n, false)
var f = Int.factorial(n-1)
f = f * f * n * size
Fmt.print("Order $d: Size $-4d x $d! x $d! => Total $d", n, size, n, n-1, f)
}</syntaxhighlight>
{{out}}
<pre>
The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! => Total 1
Order 2: Size 1 x 2! x 1! => Total 2
Order 3: Size 1 x 3! x 2! => Total 12
Order 4: Size 4 x 4! x 3! => Total 576
Order 5: Size 56 x 5! x 4! => Total 161280
Order 6: Size 9408 x 6! x 5! => Total 812851200
</pre>
=={{header|zkl}}==
{{trans|Go}}
This reuses the dList function from the [[Permutations/Derangements#zkl]] task, suitably adjusted for the present one.
<syntaxhighlight lang="zkl">fcn reducedLatinSquare(n,write=False){
if(n<=1) return(n);
rlatin:=n.pump(List(), List.createLong(n,0).copy); // matrix of zeros
foreach i in (n){ rlatin[0][i]=i+1 } // first row: (1,2,3..n)
count:=Ref(0);
// recursive closure to compute reduced latin squares and count or print them
rows,rsz := derangements(n), rows.len();
recurse:='wrap(i){
foreach r in (rsz){ // top
if(rows[r][0]!=i) continue; // filter by first column, ignore all but i
rlatin[i-1]=rows[r].copy();
foreach k,j in ([0..i-2],[1..n-1]){ // nested loop: foreach foreach
if(rlatin[k][j] == rlatin[i-1][j]){
if(r < rsz-1) continue(3); // -->top
if(i>2) return();
}
}
if(i<n) self.fcn(i + 1, vm.pasteArgs(1)); // 'wrap hides local data (ie count, rows, etc)
else{
count.inc();
if(write) printSquare(rlatin,n);
}
}
};
recurse(2); // remaining rows
return(count.value);
}
fcn derangements(n,i){
enum:=[1..n].pump(List);
Utils.Helpers.permuteW(enum).tweak('wrap(perm){
if(perm.zipWith('==,enum).sum(0)) Void.Skip
else perm
}).pump(List);
}
fcn printSquare(matrix,n){
matrix.pump(Console.println,fcn(l){ l.concat(", ","[","]") });
println();
}
fcn fact(n){ ([1..n]).reduce('*,1) }</syntaxhighlight>
<syntaxhighlight lang="zkl">println("The four reduced latin squares of order 4 are:");
reducedLatinSquare(4,True);
println("The size of the set of reduced latin squares for the following orders");
println("and hence the total number of latin squares of these orders are:");
foreach n in ([1..6]){
size,f,f := reducedLatinSquare(n), fact(n - 1), f*f*n*size;;
println("Order %d: Size %-4d x %d! x %d! -> Total %,d".fmt(n,size,n,n-1,f));
}</syntaxhighlight>
{{out}}
<pre>
The four reduced latin squares of order 4 are:
[1, 2, 3, 4]
[2, 3, 4, 1]
[3, 4, 1, 2]
[4, 1, 2, 3]
[1, 2, 3, 4]
[2, 4, 1, 3]
[3, 1, 4, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 1, 2]
[4, 3, 2, 1]
[1, 2, 3, 4]
[2, 1, 4, 3]
[3, 4, 2, 1]
[4, 3, 1, 2]
The size of the set of reduced latin squares for the following orders
and hence the total number of latin squares of these orders are:
Order 1: Size 1 x 1! x 0! -> Total 1
Order 2: Size 1 x 2! x 1! -> Total 2
Order 3: Size 1 x 3! x 2! -> Total 12
Order 4: Size 4 x 4! x 3! -> Total 576
Order 5: Size 56 x 5! x 4! -> Total 161,280
Order 6: Size 9408 x 6! x 5! -> Total 812,851,200
</pre>
| |||