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);
}