A002527 Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i and p(1) <= 3.

0, 1, 2, 6, 18, 60, 184, 560, 1695, 5200, 15956, 48916, 149664, 458048, 1402360, 4294417, 13149210, 40259178, 123260854, 377395940, 1155508592, 3537919648, 10832298239, 33165996032, 101546731816, 310913195800, 951945967120, 2914642812096, 8923975209168

Offset: 0

Author

N. J. A. Sloane

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, ones on its three subdiagonals (with the exception of a single zero in the (4,1)-entry), and is zero elsewhere.

This is the second row of Kløve's Table 3.

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 92 terms from Nathaniel Johnston)
• 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.
• R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
• Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

Programs

Maple

with(LinearAlgebra):
A002527:= n-> `if`(n=0, 0, Permanent(Matrix(n, (i, j)->
              `if`(abs(j-i)<4 and [i, j]<>[4, 1], 1, 0)))):
seq(A002527(n), n=0..20);

Mathematica

A002527[n_] := If [n == 0, 0, Permanent[Table[If [Abs[j-i]<4 && {i, j} != {4, 1}, 1, 0], {i, 1, n}, {j, 1, n}]]]; Table [A002527[n], {n, 0, 25}] (* Jean-François Alcover, Mar 11 2014, after Maple *)

Formula

From Nathaniel Johnston, Apr 03 2011: (Start)

a(n) = A002526(n) - A188379(n-1).

a(n) = a(n-1) + A002526(n-1) + A002529(n-1). (End)

G.f.: x*(x^7+2*x^6-2*x^4-2*x^3-1) / (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 07 2011

Extensions

Name and comments edited, and terms after a(11) added by Nathaniel Johnston, Apr 03 2011