A173103 The number of possible borders of Latin squares with the top row fixed.
1, 1, 2, 26, 924, 81624, 13433520, 3706068240, 1582042381920, 987057348842880, 861632512758823680, 1016677874552767660800, 1576819957670934809817600, 3140963381712726319842892800, 7880571655922780897709237811200, 24492587962448960350527019884595200
Offset: 1
Keywords
Examples
The only two configurations for n=3, given the top row is 123: 123 123 2 1 3 2 312 231 Two arbitrary configurations for n=4, given the top row is 1234: 1234 1234 2 1 4 3 3 2 3 2 4123 2341
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.
Programs
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Maple
d:= proc(n) d(n):= `if`(n<=1, 1-n, (n-1)*(d(n-1)+d(n-2))) end: a:= proc(n) a(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: seq(a(n), n=1..20); # Alois P. Heinz, Aug 18 2013 -
Mathematica
d = Subfactorial; a[n_] := If[n <= 3, {1, 1, 2}[[n]], (n-2)! (((2n-5) d[n-2]^2)/(n-3) + 2d[n-1] d[n-2] + ((n-1) d[n-1]^2)/(n-2))]; Array[a, 20] (* Jean-François Alcover, Nov 10 2020 *)
Formula
For n>3, a(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).
Comments