A335088 -id:A335088 - cp's OEIS frontend

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

Geoffrey Critzer, Jun 05 2013

nonn

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

			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.

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

Cf. A000142, A079128, A335088.

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

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}]

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

A335797 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k!).

Original entry on oeis.org

1, 1, 1, 4, 5, 51, 7, 876, 457, 7678, 5271, 678569, 10705, 27644436, 5060161, 133924576, 197920145, 82864869803, 173283535, 5832742205056, 98269310261, 34660429169122, 25313714237505, 44152005855084345, 13685698802401, 2410161938206898126, 129066382491033573

Offset: 0

Ilya Gutkovskiy, Oct 12 2020

nonn

Number of set partitions of [n] into blocks that are relatively prime to n.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A057562, A275429, A335088.

b:= proc(n, m) option remember; `if`(n=0, 1, add(`if`(
      igcd(j, m)=1, b(n-j, m), 0)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..27);  # Alois P. Heinz, Oct 12 2020

Table[n! SeriesCoefficient[Exp[Sum[Boole[GCD[n, k] == 1] x^k/k!, {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

cp's OEIS Frontend

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

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