A381461 Number of permutations of [n] with no fixed points where adjacent elements differ by at least 3.

1, 0, 0, 0, 0, 0, 2, 8, 115, 1274, 15099, 179628, 2260064, 30534802, 441269110, 6789665680, 110947884520, 1920180939650, 35099424286573, 675866037989156, 13676799446869485, 290208293166279344, 6443880771921767240

Offset: 0

Author

Alois P. Heinz, Feb 24 2025

Keywords

• nonn
• more

Examples

			a(6) = 2: 362514, 415263.
a(7) = 8: 2516374, 3615274, 3625174, 3627415, 3741625, 4152736, 4163725, 4173625.
a(8) = 115: 25163847, 25174836, 25184736, 25814736, ..., 84736251, 85263714, 85263741, 85274163.
		

Links

• Robert P. P. McKone, The permutations n=6 to n=11.
• Wikipedia, Permutation

Crossrefs

Cf. A000166, A002464, A127697, A288208.

Programs

Maple

b:= proc(s, l) option remember; (n-> `if`(n=0, 1, add(
     `if`(j=n or abs(l-j)<3, 0, b(s minus {j}, j)), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, -2):
seq(a(n), n=0..16);

Mathematica

Clear[permCount]; permCount[s_, last_] := permCount[s, last] = Module[{n, j}, n = Length[s]; If[n == 0, 1, Total[Table[If[j == n || Abs[last - j] < 3, 0, permCount[Complement[s, {j}], j]], {j, s}]]]]; Table[permCount[Range[n], -2], {n, 0, 12}] (* Robert P. P. McKone, Mar 01 2025 *)