A217447 Number of n x n permutation matrices that disconnect their zeros.

2, 6, 12, 32, 120, 580, 3392, 23244, 182776, 1622468, 16045200, 174894172, 2082824744, 26902998516, 374570250688, 5591767768460, 89095070783832, 1509041577895204, 27073887615758576, 512898265609845948, 10230945527263709320, 214337863242231108692

Offset: 2

Author

Brian Hopkins, Nov 16 2012

Keywords

• easy
• nonn

Examples

			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.
		

Links

• D. Edelman and A. Edelman, Solution 1877: Disconnecting a permutation matrix, Math. Mag. 85 (2012) 297-298.

Crossrefs

Terms from A007489 in formula.

Programs

Mathematica

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 *)

Formula

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.

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

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

a(n) ~ 4 * (n-2)!. - Vaclav Kotesovec, Jan 31 2014