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

Author

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

Keywords

• nonn

Links

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

Crossrefs

Cf. A000142, A088994.

Programs

Maple

b:= proc(n, i) option remember; `if`(((i+1)/2)^2n, 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 *)

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

Formula

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

Extensions

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