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. (Formerly M1626 N0637)

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

0,3

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).

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

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 *)

Crossrefs

Sequence in context: A148460 A148461 A358251 * A005566 A005631 A118677

Adjacent sequences: A002524 A002525 A002526 * A002528 A002529 A002530

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved