A037224 Number of permutations p of {1,2,3...,n} that are fixed points under the operation of first reversing p, then taking the inverse.

1, 0, 0, 2, 2, 0, 0, 12, 12, 0, 0, 120, 120, 0, 0, 1680, 1680, 0, 0, 30240, 30240, 0, 0, 665280, 665280, 0, 0, 17297280, 17297280, 0, 0, 518918400, 518918400, 0, 0, 17643225600, 17643225600, 0, 0, 670442572800, 670442572800, 0, 0, 28158588057600

Offset

1,4

Comments

A122670 is an essentially identical sequence.

Also the number of rotationally symmetric solutions to non-attacking rooks problem on an n X n board.

Reversal of a permutation reflects the associated permutation matrix through an axis parallel to its sides, while inversion reflects the matrix through its main diagonal. The composition of these two operations is rotation by 90 degrees, and so permutations fixed by this composition correspond to rotationally symmetric rook diagrams by taking the associated permutation matrix. [Ian Duff, Mar 09 2007 and Joel B. Lewis, Jun 10 2009]

Equivalently, the number of permutations fixed by first inverting and then reversing. We may also replace "reversing" with "complementing" in the preceding sentences, where the complement of (w(1), ..., w(n)) is (n + 1 - w(1), ..., n + 1 - w(n)). [Joel B. Lewis, Jun 10 2009]

Links

Table of n, a(n) for n=1..44.

C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.

M. Szabo, Non-attacking Queens Problem Page

Formula

a(4n) = a(4n+1) = (2n-1)!*2/(n-1)!, a(4n+2) = a(4n+3) = 0.

Example

Let p be the permutation {11,1,9,3,7,5,8,6,10,4,12,2} of {1,2,3,..,12}. Then the reverse Rp of p is {2,12,4,10,6,8,5,7,3,9,1,11} and the inverse IRp of Rp is {11,1,9,3,7,5,8,6,10,4,12,2}. Thus p counts as one of the a(12)=120 fixed-points for n=12.

Maple

a:= n-> `if`(irem(n, 4, 'm')>1, 0,
        `if`(m=0, 1, (2*m-1)! * 2/(m-1)!)):
seq(a(n), n=1..99);

Mathematica

{1}~Join~Table[If[MemberQ[{0, 1}, Mod[n, 4]], (2 # - 1)!*2/(# - 1)! &[Floor[n/4]], 0], {n, 2, 44}] (* Michael De Vlieger, Oct 05 2016 *)

Prog

(PARI)
a(n)=
{
    if ( n%4>=2, return(0) );
    n = n\4;
    if ( n==0, return(1) );
    return( (2*n-1)!*2/(n-1)! );
}
vector(55, n, a(n)) /* Joerg Arndt, Jan 21 2011 */

Crossrefs

Cf. A001813, A033148, A032522, A037223, A122670.

Sequence in context: A282699 A182107 A337999 * A122670 A352661 A283494

Adjacent sequences: A037221 A037222 A037223 * A037225 A037226 A037227

Keyword

nonn,easy

Author

Miklos SZABO (mike(AT)ludens.elte.hu)

Extensions

Edited by N. J. A. Sloane, Jun 12 2009, incorporating comments from John W. Layman, Sep 17 2004

Status

approved