A075852 Number of permutations s of {1,2,...,n} such that |s(i)-i|>3 for each i=1,2,...,n.

1, 0, 0, 0, 0, 0, 0, 0, 1, 16, 436, 6984, 114124, 1799688, 29125117, 486980182, 8490078104, 154750897552, 2951968964768, 58917663227568, 1229367602071416, 26787823838035750, 608794318333169289, 14411810690642972432

Offset: 0

Author

Reiner Martin, Oct 15 2002

Keywords

• nonn

Comments

a(n) equals the permanent of the n X n matrix with 0's along the main diagonal, the subdiagonal, the superdiagonal, the sub-subdiagonal, the super-superdiagonal, the sub-sub-subdiagonal, the super-super-superdiagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• George Spahn and Doron Zeilberger, Automatic Counting of Generalized Latin Rectangles and Trapezoids, arXiv:2108.11285 [math.CO], 2021.

Crossrefs

Cf. A001883, A075851.

Programs

Maple

b:= proc(s) option remember; (n-> `if`(n=0, 1, add(
      `if`(abs(n-i)>3, b(s minus {i}), 0), i=s)))(nops(s))
    end:
a:= n-> b({$1..n}):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 25 2019

Mathematica

a[0] = 1; a[n_] := a[n] = If[n < 8, 0, SparseArray[{Band[{1, 1}] -> 0, Band[{2, 1}] -> 0, Band[{3, 1}] -> 0, Band[{4, 1}] -> 0, Band[{1, 2}] -> 0, Band[{1, 3}] -> 0, Band[{1, 4}] -> 0}, {n, n}, 1] // Permanent];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, May 01 2019 *)

Extensions

More terms from Vladimir Baltic, Vladeta Jovovic, Jan 04 2003

a(21) from Alois P. Heinz, Jul 04 2015

a(22)-a(23) from Alois P. Heinz, Jan 22 2019

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019