cp's OEIS Frontend

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.

%I A037224 #36 Jun 14 2024 08:24:52
%S A037224 1,0,0,2,2,0,0,12,12,0,0,120,120,0,0,1680,1680,0,0,30240,30240,0,0,
%T A037224 665280,665280,0,0,17297280,17297280,0,0,518918400,518918400,0,0,
%U A037224 17643225600,17643225600,0,0,670442572800,670442572800,0,0,28158588057600
%N 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.
%C A037224 A122670 is an essentially identical sequence.
%C A037224 Also the number of rotationally symmetric solutions to non-attacking rooks problem on an n X n board.
%C A037224 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]
%C A037224 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]
%H A037224 C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="https://arXiv.org/abs/math.CO/0506334">On the X-rays of permutations</a>, arXiv:math/0506334 [math.CO], 2005.
%H A037224 M. Szabo, <a href="http://www.nexus.hu/mikk/queen/index.html">Non-attacking Queens Problem Page</a>
%F A037224 a(4n) = a(4n+1) = (2n-1)!*2/(n-1)!, a(4n+2) = a(4n+3) = 0.
%e A037224 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.
%p A037224 a:= n-> `if`(irem(n, 4, 'm')>1, 0,
%p A037224         `if`(m=0, 1, (2*m-1)! * 2/(m-1)!)):
%p A037224 seq(a(n), n=1..99);  # _Alois P. Heinz_, Jan 21 2011
%t A037224 {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 *)
%o A037224 (PARI)
%o A037224 a(n)=
%o A037224 {
%o A037224     if ( n%4>=2, return(0) );
%o A037224     n = n\4;
%o A037224     if ( n==0, return(1) );
%o A037224     return( (2*n-1)!*2/(n-1)! );
%o A037224 }
%o A037224 vector(55,n,a(n)) /* _Joerg Arndt_, Jan 21 2011 */
%Y A037224 Cf. A001813, A033148, A032522, A037223, A122670.
%K A037224 nonn,easy
%O A037224 1,4
%A A037224 Miklos SZABO (mike(AT)ludens.elte.hu)
%E A037224 Edited by _N. J. A. Sloane_, Jun 12 2009, incorporating comments from _John W. Layman_, Sep 17 2004