A173104 The number of possible borders of Latin squares.

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

Offset

1,2

Comments

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

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.

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).

Example

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

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 *)

Crossrefs

Related to A000166. Equals A173103 multiplied by n!.

Sequence in context: A108078 A052129 A216335 * A141770 A363098 A230265

Adjacent sequences: A173101 A173102 A173103 * A173105 A173106 A173107

Keyword

nonn

Author

Johan de Ruiter, Feb 09 2010

Status

approved