A226388 Number of n-permutations such that all cycle lengths have a common divisor >= 2.

0, 0, 1, 2, 9, 24, 265, 720, 11025, 62720, 965601, 3628800, 130478425, 479001600, 19151042625, 191132125184, 4108830350625, 20922789888000, 1448301616386625, 6402373705728000, 466136852576275881, 5675242696048640000, 193688172394325870625, 1124000727777607680000

Offset: 0

Author

Geoffrey Critzer, Jun 05 2013

Keywords

• nonn

Comments

a(p) = (p-1)! for p a prime.

Examples

			a(6) = 265 counting permutations with cycle types: 6; 4-2; 3-3; 2-2-2; of which there are 120 + 90 + 40 + 15 = 265.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000142, A079128, A335088.

Programs

Maple

with(combinat):
b:= proc(n, i, g) option remember; `if`(n=0, `if`(g>1, 1, 0),
      `if`(i<2, 0, b(n, i-1, g) +`if`(igcd(g, i)<2, 0,
       add((i-1)!^j/j! *multinomial(n, i$j, n-i*j)*
         b(n-i*j, i-1, igcd(i, g)), j=1..n/i))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 06 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(g>1, 1, 0), add(
      (j-1)!*b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 04 2021

Mathematica

f[list_] :=
Total[list]!/Apply[Times, Table[list[[i]], {i, 1, Length[list]}]]/
  Apply[Times,
   Select[Table[
      Count[list, i], {i, 1, Total[list]}], # > 0 &]!]; Table[
Total[Map[f, Select[Partitions[n], Apply[GCD, #] > 1 &]]], {n, 0,
  25}]

Formula

a(n) = n! - A079128(n) for n >= 1. - Alois P. Heinz, Jul 04 2021