A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

1, 1, 0, 2, 8, 24, 144, 720, 8448, 64512, 576000, 5529600, 74972160, 887546880, 11285084160, 168318259200, 2843121254400, 44790578380800, 747955947110400, 13937735643955200, 287117441217331200, 5838778006909747200, 120976472421826560000, 2712639152754878054400

Offset: 0

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003

Keywords

• nonn

Comments

a(n) is the number of n-permutations composed only of odd cycles of distinct length. - Geoffrey Critzer, Mar 08 2013

Also the number of permutations p of [n] with unique (functional) square root, i.e., there exists a unique permutation g such that g^2 = p. - Keith J. Bauer, Jan 08 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000142, A007838, A007841, A087639, A088335, A294506, A305199, A309319.

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-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2017

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[Product[1+x^(2i-1)/(2i-1),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Mar 08 2013 *)

PARI

{a(n)=n!*polcoeff( prod(k=1, n, 1+(k%2)*x^k/k, 1+x*O(x^n)), n)} /* Michael Somos, Sep 19 2006 */

Formula

E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - Vladeta Jovovic, Nov 05 2003

a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

a(n) = n! - A088335(n). - Alois P. Heinz, Jan 27 2020

Extensions

More terms from Vladeta Jovovic, Nov 03 2003

a(0)=1 prepended by Seiichi Manyama, Nov 01 2017