A188498 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, and p(j) >= 2 for j=3,4.

0, 1, 2, 3, 8, 30, 102, 308, 905, 2744, 8473, 26112, 79924, 244204, 747160, 2288521, 7009458, 21461803, 65704200, 201162258, 615922714, 1885853660, 5774072225, 17678809840, 54128358209, 165728860112, 507424764216, 1553620027784, 4756831354752

Offset: 0

Author

N. J. A. Sloane, Apr 01 2011

Keywords

• nonn

Comments

a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of zeros in the (1,3) and (1,4)-entries), ones on its three subdiagonals (with the exception of zeros in the (3,1) and (4,1)-entries), and is zero elsewhere.

This is row 13 of Kløve's Table 3.

Links

• Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Programs

Maple

with(LinearAlgebra):
A188498:= n-> `if` (n=0, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[1, 3] and [i, j]<>[1, 4] and [i, j]<>[3, 1] and [i, j]<>[4, 1], 1, 0)))):
seq (A188498(n), n=0..20);

Mathematica

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {1, 3} && {i, j} != {1, 4} && {i, j} != {3, 1} && {i, j} != {4, 1}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 07 2016, adapted from Maple *)
CoefficientList[Series[-(x^10 + 2 x^9 + x^8 - 2 x^6 - 2 x^5 - 2 x^4 - 3 x^3 + x) / (x^14 + 2 x^13 + 2 x^11 + 4 x^10 - 2 x^9 - 10 x^8 - 16 x^7 - 2 x^6 + 8 x^5 + 10 x^4 + 2 x^2 + 2 x - 1), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 07 2016 *)

PARI

concat(0, Vec(-(x^10+2*x^9+x^8 -2*x^6-2*x^5-2*x^4 -3*x^3+x) / (x^14+2*x^13+2*x^11 +4*x^10-2*x^9-10*x^8 -16*x^7-2*x^6+8*x^5 +10*x^4+2*x^2+2*x-1) + O(x^40))) \\ Michel Marcus, Dec 12 2014

Formula

From Nathaniel Johnston, Apr 11 2011: (Start)

a(n) = A188497(n+1) - A188494(n).

a(n) = A002526(n-1) + A002526(n-2).

(End)

G.f.: -(x^10 + 2*x^9 + x^8 - 2*x^6 - 2*x^5 - 2*x^4 - 3*x^3 + x) / (x^14 + 2*x^13 + 2*x^11 + 4*x^10 - 2*x^9 - 10*x^8 - 16*x^7 - 2*x^6 + 8*x^5 + 10*x^4 + 2*x^2 + 2*x - 1).

Extensions

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 11 2011