A226388 - OEIS

%I #23 Jul 04 2021 15:02:30

%S 0,0,1,2,9,24,265,720,11025,62720,965601,3628800,130478425,479001600,

%T 19151042625,191132125184,4108830350625,20922789888000,

%U 1448301616386625,6402373705728000,466136852576275881,5675242696048640000,193688172394325870625,1124000727777607680000

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

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

%H Alois P. Heinz, <a href="/A226388/b226388.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = n! - A079128(n) for n >= 1. - _Alois P. Heinz_, Jul 04 2021

%e 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.

%p with(combinat):

%p b:= proc(n, i, g) option remember; `if`(n=0, `if`(g>1, 1, 0),

%p       `if`(i<2, 0, b(n, i-1, g) +`if`(igcd(g, i)<2, 0,

%p        add((i-1)!^j/j! *multinomial(n, i$j, n-i*j)*

%p          b(n-i*j, i-1, igcd(i, g)), j=1..n/i))))

%p     end:

%p a:= n-> b(n, n, 0):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 06 2013

%p # second Maple program:

%p b:= proc(n, g) option remember; `if`(n=0, `if`(g>1, 1, 0), add(

%p       (j-1)!*b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 04 2021

%t f[list_] :=

%t Total[list]!/Apply[Times, Table[list[[i]], {i, 1, Length[list]}]]/

%t   Apply[Times,

%t    Select[Table[

%t       Count[list, i], {i, 1, Total[list]}], # > 0 &]!]; Table[

%t Total[Map[f, Select[Partitions[n], Apply[GCD, #] > 1 &]]], {n, 0,

%t   25}]

%Y Cf. A000142, A079128, A335088.

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Jun 05 2013