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A173103 The number of possible borders of Latin squares with the top row fixed.

%I A173103 #25 Nov 10 2020 15:44:31
%S A173103 1,1,2,26,924,81624,13433520,3706068240,1582042381920,987057348842880,
%T A173103 861632512758823680,1016677874552767660800,1576819957670934809817600,
%U A173103 3140963381712726319842892800,7880571655922780897709237811200,24492587962448960350527019884595200
%N A173103 The number of possible borders of Latin squares with the top row fixed.
%C A173103 The definition is not quite right, and should be corrected.
%H A173103 Alois P. Heinz, <a href="/A173103/b173103.txt">Table of n, a(n) for n = 1..100</a>
%H A173103 J. de Ruiter, <a href="http://liacs.leidenuniv.nl/assets/Bachelorscripties/10-04-JohandeRuiter.pdf">On Jigsaw Sudoku Puzzles and Related Topics</a>, Bachelor Thesis, Leiden Institute of Advanced Computer Science, 2010.
%F A173103 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).
%e A173103 The only two configurations for n=3, given the top row is 123:
%e A173103   123   123
%e A173103   2 1   3 2
%e A173103   312   231
%e A173103 Two arbitrary configurations for n=4, given the top row is 1234:
%e A173103   1234   1234
%e A173103   2  1   4  3
%e A173103   3  2   3  2
%e A173103   4123   2341
%p A173103 d:= proc(n) d(n):= `if`(n<=1, 1-n, (n-1)*(d(n-1)+d(n-2))) end:
%p A173103 a:= proc(n) a(n):= `if`(n<4, [1, 1, 2][n], (n-2)!*((n-1)/
%p A173103        (n-2)*d(n-1)^2+2*d(n-1)*d(n-2)+(2*n-5)/(n-3)*d(n-2)^2))
%p A173103     end:
%p A173103 seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 18 2013
%t A173103 d = Subfactorial;
%t A173103 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))];
%t A173103 Array[a, 20] (* _Jean-François Alcover_, Nov 10 2020 *)
%Y A173103 Related to A000166. Equals A173104 divided by n!.
%K A173103 nonn
%O A173103 1,3
%A A173103 _Johan de Ruiter_, Feb 09 2010