A372579 Number of permutations of [n] such that the number of cycles of length k is zero or equals k for every k.

1, 1, 0, 0, 3, 15, 0, 0, 0, 2240, 22400, 0, 0, 4804800, 67267200, 0, 3405402000, 57891834000, 0, 0, 49497518070000, 1039447879470000, 0, 0, 0, 56947245360343962624, 1480628379368943028224, 0, 0, 4057662073660588368847872, 121729862209817651065436160, 0, 0, 0

Offset: 0

Author

Alois P. Heinz, Jul 04 2024

Keywords

• nonn

Examples

			a(5) = 15 = 5*3: (1)(23)(45), (1)(24)(35), (1)(25)(34), ..., (1,2)(3,4)(5),
  (1,3)(2,4)(5), (1,4)(2,3)(5).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..470
• Wikipedia, Permutation

Crossrefs

Cf. A001422, A003995, A372545, A374262, A374292, A374321.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or j=i, b(n-i*j, i-1)*(i-1)!^j/j!*
      combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

Formula

a(n) = 0 <=> n in { A001422 }.

a(n) > 0 <=> n in { A003995 }.