A088335 Number of permutations in the symmetric group S_n such that the size of their centralizer is even.

0, 0, 2, 4, 16, 96, 576, 4320, 31872, 298368, 3052800, 34387200, 404029440, 5339473920, 75893207040, 1139356108800, 18079668633600, 310896849715200, 5654417758617600, 107707364764876800, 2145784566959308800, 45252164164799692800, 1003024255355781120000

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(n) = n! - A088994(n).

Maple

b:= proc(n, i) option remember; `if`(((i+1)/2)^2<n, 0,
      `if`(n=0, 1, b(n, i-2)+`if`(i>n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> n!-b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 27 2020

Mathematica

b[n_, i_] := b[n, i] = If[((i + 1)/2)^2 < n, 0, If[n == 0, 1, b[n, i - 2] + If[i > n, 0, (i - 1)! b[n - i, i - 2] Binomial[n, i]]]];
a[n_] := n! - b[n, n - 1 + Mod[n, 2]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 08 2020, after Alois P. Heinz *)

Prog

(PARI) seq(n)={Vec(serlaplace(1/(1-x) - prod(k=1, n, 1+(k%2)*x^k/k + O(x*x^n))), -(n+1))} \\ Andrew Howroyd, Jan 27 2020

Crossrefs

Cf. A000142, A088994.

Sequence in context: A052835 A009565 A009838 * A066318 A308606 A066952

Adjacent sequences: A088332 A088333 A088334 * A088336 A088337 A088338

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 07 2003

Extensions

a(0)=0 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 27 2020

Status

approved