

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.


8



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
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OFFSET

1,4


COMMENTS

A122670 is an essentially identical sequence.
Also the number of rotationally symmetric solutions to nonattacking 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 xrays of permutations, arXiv:math/0506334 [math.CO], 2005.
M. Szabo, Nonattacking Queens Problem Page


FORMULA

a(4n) = a(4n+1) = (2n1)!*2/(n1)!, 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 fixedpoints for n=12.


MAPLE

a:= n> `if`(irem(n, 4, 'm')>1, 0,
`if`(m=0, 1, (2*m1)! * 2/(m1)!)):
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*n1)!*2/(n1)! );
}
vector(55, n, a(n)) /* Joerg Arndt, Jan 21 2011 */


CROSSREFS

Cf. A001813, A033148, A032522, A037223, A122670.
Sequence in context: A230275 A230592 A182107 * A122670 A190389 A176127
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



