A173104 The number of possible borders of Latin squares.

1, 2, 12, 624, 110880, 58769280, 67704940800, 149428671436800, 574091539551129600, 3581833707481042944000, 34393612685291413069824000, 486990328595374993951457280000, 9818890674272030616178239406080000, 273823820339488809857168046768783360000

Offset: 1

Author

Johan de Ruiter, Feb 09 2010

Keywords

• nonn

Comments

The definition is not quite right, and should be corrected.

Examples

			Two arbitrary configurations for n=3:
  123   312
  2 1   1 3
  312   231
Two arbitrary configurations for n=4:
  1234   1432
  2  1   3  4
  3  2   4  1
  4123   2143
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100
• J. de Ruiter, On Jigsaw Sudoku Puzzles and Related Topics, Bachelor Thesis, Leiden Institute of Advanced Computer Science, 2010.

Crossrefs

Related to A000166. Equals A173103 multiplied by n!.

Programs

Maple

d:= proc(n) d(n):= `if`(n<=1, 1-n, (n-1)*(d(n-1)+d(n-2))) end:
b:= proc(n) b(n):= `if`(n<4, [1, 1, 2][n], (n-2)!*((n-1)/
       (n-2)*d(n-1)^2+2*d(n-1)*d(n-2)+(2*n-5)/(n-3)*d(n-2)^2))
    end:
a:= n-> n!*b(n):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 18 2013

Mathematica

d = Subfactorial;
a[n_] := If[n <= 3, {1, 2, 12}[[n]], n! (n-2)! ((n-1)/(n-2) d[n-1]^2 + 2d[n-1] d[n-2] + (2n-5)/(n-3) d[n-2]^2)];
Array[a, 20] (* Jean-François Alcover, Nov 10 2020 *)

Formula

For n>3, a(n)=n!(n-2)!((n-1)/(n-2)d[n-1]^2+2d[n-1]d[n-2]+(2n-5)/(n-3)d[n-2]^2), where d[k] is the number of derangements of k elements (A000166).