A188493 a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).

0, 0, 2, 6, 14, 31, 104, 344, 1084, 3236, 9784, 29964, 92241, 282780, 865064, 2646292, 8102454, 24813838, 75982346, 232630527, 712230076, 2180675264, 6676819512, 20443032008, 62591840320, 191641545768, 586762729889, 1796535598952, 5500587026592

Offset: 0

Author

N. J. A. Sloane, Apr 01 2011

Keywords

• nonn

Comments

For n >= 2, a(n) is the number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(j) <= 2+j for j = 1,2, and p(j) >= j-2 for j = 4,5.

For n >= 2, 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 (1,4) and (2,5)-entries), ones on its three subdiagonals (with the exception of zeros in the (4,1) and (5,2)-entries), and is zero elsewhere.

This is row 7 of Kløve's Table 3.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..124
• 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):
A188493:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[4, 1] and [i, j]<>[5, 2] and [i, j]<>[1, 4] and [i, j]<>[2, 5], 1, 0)))):
seq (A188493(n), n=0..20);

Mathematica

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {4, 1} && {i, j} != {5, 2} && {i, j} != {1, 4} && {i, j} != {2, 5}, 1, 0], {i, 1, n}, {j, 1, n}] ]; a[1] = 0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)

Formula

G.f.: -(x^10+2*x^9+2*x^7 +4*x^6-2*x^5-6*x^4 -9*x^3-2*x^2+2*x+2) *x^2 / (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). - Alois P. Heinz, Apr 08 2011

Extensions

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 08 2011