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A037224
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Number of permutations p of {1,2,3...,n} that are fixed points under the operation of first reversing p, then taking the inverse.
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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. [From Ian Duff (ianfduff(AT)yahoo.co.uk), Mar 09 2007 and Joel Brewster Lewis (jblewis(AT)post.harvard.edu), 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)). [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), Jun 10 2009]
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LINKS
| C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the x-rays of permutations
M. Szabo, Non-attacking Queens Problem Page
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FORMULA
| a(4n) = a(4n+1) = (2n-1)!*2/(n-1)!, a(4n+2) = a(4n+3) = 0.
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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.
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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);
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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)) /* show terms */ /* Joerg Arndt, Jan 21 2011 */
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CROSSREFS
| Cf. A001813, A033148, A032522, A037223.
Sequence in context: A069971 A167291 A063865 * A122670 A190389 A176127
Adjacent sequences: A037221 A037222 A037223 * A037225 A037226 A037227
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KEYWORD
| nonn,easy
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AUTHOR
| Miklos SZABO (mike(AT)ludens.elte.hu)
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EXTENSIONS
| Edited by N. J. A. Sloane, Jun 12 2009, incorporating comments from John W. Layman (layman(AT)math.vt.edu), Sep 17 2004
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