A362362 Number of permutations of [n] such that each cycle contains its length as an element.

1, 1, 1, 3, 8, 36, 174, 1104, 7440, 62640, 545040, 5649840, 60681600, 748621440, 9518342400, 136758585600, 2009451628800, 32848492723200, 549241915622400, 10066913176320000, 188293339922688000, 3832031198451456000, 79291640831090688000, 1771146970953744384000

Offset

0,4

Comments

The cycle lengths are distinct as a consequence of the definition.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Wikipedia, Permutation

Example

a(3) = 3: (123), (132), (1)(23).
a(4) = 8: (1234), (1243), (1324), (1342), (1423), (1432), (1)(234), (1)(243).

Maple

a:= n-> add((n-nops(p))!, p=select(l-> nops(l)=
        nops({l[]}), combinat[partition](n))):
seq(a(n), n=0..24);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, p!, b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..24);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[i*(i + 1)/2 < n, 0, If[n == 0, p!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 15 2023, from second Maple program *)

Crossrefs

Cf. A000009, A007838, A032020, A179973, A321520, A326493, A364277, A364406.

Sequence in context: A087905 A295103 A299329 * A020111 A367640 A111543

Adjacent sequences: A362359 A362360 A362361 * A362363 A362364 A362365

Keyword

nonn

Author

Alois P. Heinz, Jul 05 2023

Status

approved