A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.

1, 1, 0, 1, 0, 0, 6, 6, 0, 0, 720, 2160, 9360, 19440, 30240, 3659040, 21772800, 228614400, 1632960000, 11125900800, 73025971200, 1708337433600, 15442053580800, 254260755302400, 3318429200486400, 46929444097536000, 546974781889536000, 7312714579602432000

Offset: 0

Author

Alois P. Heinz, Jul 22 2023

Keywords

• nonn

Examples

			a(0) = 1: () the empty permutation.
a(1) = 1: (1).
a(3) = 1: (1)(23).
a(6) = 6: (1)(24)(356), (1)(24)(365), (1)(25)(346), (1)(25)(364),
  (1)(26)(345), (1)(26)(354).
a(7) = 6: (1)(23)(4567), (1)(23)(4576), (1)(23)(4657), (1)(23)(4675),
  (1)(23)(4756), (1)(23)(4765).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..474
• Wikipedia, Permutation

Crossrefs

Cf. A000142, A007837, A188431, A362362, A364207, A364277 A364279 A364281 A364283.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n+1, 0, b(n-i, i-1)*binomial(n-i, i-1)*(i-1)!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);

Mathematica

b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[2*i > n + 1, 0, b[n - i, i - 1]*Binomial[n - i, i - 1]*(i - 1)!]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)