A346121 Number of permutations of [n] whose order is a multiple of n.

1, 1, 2, 6, 24, 240, 720, 5040, 40320, 816480, 3628800, 108108000, 479001600, 14789174400, 254431457280, 1307674368000, 20922789888000, 872545722048000, 6402373705728000, 411616608508385280, 7817896752906240000, 128126503414990080000, 1124000727777607680000

Offset: 1

Author

Alois P. Heinz, Jul 05 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..450
• Wikipedia, Permutation

Crossrefs

Cf. A000142, A000961, A052699, A057731, A074351, A074759.

Programs

Maple

b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> (p-> add(coeff(p, x, i*n), i=1..degree(p)/n))(b(n, 1)):
seq(a(n), n=1..23);
# second Maple program:
h:= proc(n, j) option remember; uses padic, numtheory; n/mul(`if`(
      ordp(j, p) b(n$2):
seq(a(n), n=1..23);

Mathematica

b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*b[n - j, LCM[g, j]]* Binomial[n - 1, j - 1], {j, 1, n}]] // Expand;
a[n_] := With[{p = b[n, 1]}, Sum[Coefficient[p, x, i*n], {i, 1, Exponent[p, x]/n}]];
Array[a, 40] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz's first program *)

Formula

a(n) = Sum_{i>=1} A057731(n,i*n).

a(n) = (n-1)! <=> n in { A000961 }.

a(n) = A057731(n,n) = A074351(n) = A052699(n-1) for n <= 9.