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

0,5

Links

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

Wikipedia, Permutation

Formula

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

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

Example

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).

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);

Crossrefs

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

Sequence in context: A048568 A119678 A374321 * A096682 A371099 A331228

Adjacent sequences: A372576 A372577 A372578 * A372580 A372581 A372582

Keyword

nonn,new

Author

Alois P. Heinz, Jul 04 2024

Status

approved