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%I #20 Sep 18 2017 10:06:56
%S 2,6,12,32,120,580,3392,23244,182776,1622468,16045200,174894172,
%T 2082824744,26902998516,374570250688,5591767768460,89095070783832,
%U 1509041577895204,27073887615758576,512898265609845948,10230945527263709320,214337863242231108692
%N Number of n x n permutation matrices that disconnect their zeros.
%H D. Edelman and A. Edelman, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.4.295">Solution 1877: Disconnecting a permutation matrix</a>, Math. Mag. 85 (2012) 297-298.
%F a(n) = 4 Sum_{i=1..n-2} i! - 2(n-2) Sum_{i=0..n-4} i! + 2 Sum_{i=1..n-3} i! + 2.
%F Conjecture: 2*a(n) + 2*(-n-1)*a(n-1) + (6*n-11)*a(n-2) + (-5*n+14)*a(n-3) + 3*a(n-4) + (n-6)*a(n-5) = 0. - _R. J. Mathar_, Nov 30 2012
%F Recurrence (for n>4): (2*n^2 - 16*n + 31)*a(n) = (2*n^3 - 16*n^2 + 33*n - 6)*a(n-1) - (2*n-7)*(2*n^2 - 12*n + 15)*a(n-2) + (n-4)*(2*n^2 - 12*n + 17)*a(n-3). - _Vaclav Kotesovec_, Jan 31 2014
%F a(n) ~ 4 * (n-2)!. - _Vaclav Kotesovec_, Jan 31 2014
%e The matrix corresponding to {4,3,1,2} disconnects its zeros since the 0 in the bottom left is not horizontally or vertically adjacent to another 0. In contrast, the matrix corresponding to {4,2,1,3} connects its zeros.
%t Table[4*Sum[i!, {i, n - 2}] - 2*(n - 2)*Sum[i!, {i, 0, n - 4}] + 2*Sum[i!, {i, n - 3}] + 2, {n, 2, 25}] (* _T. D. Noe_, Nov 16 2012 *)
%Y Terms from A007489 in formula.
%K easy,nonn
%O 2,1
%A _Brian Hopkins_, Nov 16 2012
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