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%I #20 Jun 07 2021 14:09:33
%S 1,2,0,0,2,3,2,5,10,17,22,44,68,127,184,356,530,1017,1502,2906,4312,
%T 8351,12388,24067,35748,69577,103404,201642,299882,585691,871498,
%U 1704509,2537522,4969153,7400782,14508938,21617096,42422023,63226948,124191257,185155568,363985681,542815792,1067892398,1592969006
%N The number of 123-avoiding simple involutions of length n.
%C An interval in a permutation is a set of contiguous indices such that the set of values of these indices under the permutation is also contiguous. A permutation is simple if it has no proper intervals (those of length more than 1 and less than the whole permutation). - _Charles R Greathouse IV_, Nov 06 2013
%H Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vincent Vatter, <a href="http://arxiv.org/abs/1310.7003">Pattern-avoiding involutions: exact and asymptotic enumeration</a>, arxiv:1310.7003, 2013.
%F G.f.: x*(-1-2*x+x^6+2*x^3+6*x^5+2*x^7+4*x^5*(-3*x^4-2*x^2+1)^(1/2)+2*x^7*(-3*x^4-2*x^2+1)^(1/2)+x^4*(-3*x^4-2*x^2+1)^(1/2)+2*x^6*(-3*x^4-2*x^2+1)^(1/2)-2*x*(-3*x^4-2*x^2+1)^(1/2)-(-3*x^4-2*x^2+1)^(1/2)+x^2+3*x^4)/(3*x^6+2*x^6*(-3*x^4-2*x^2+1)^(1/2)+5*x^4+3*x^4*(-3*x^4-2*x^2+1)^(1/2)+x^2-1-(-3*x^4-2*x^2+1)^(1/2)).
%F a(n) ~ (2*sqrt(3)+3 + (-1)^n*(2*sqrt(3)-3)) * 3^(n/2) / (12 * sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Jan 27 2015
%e a(8) = 5 because there are 5 simple involutions of length 8 which avoid the pattern 123: 58371642, 64827153, 68375142, 75382614, and 75842613.
%o (PARI) x='x+O('x^66); Vec((-1-2*x+x^6+2*x^3+6*x^5+2*x^7+4*x^5*(-3*x^4-2*x^2+1)^(1/2)+2*x^7*(-3*x^4-2*x^2+1)^(1/2)+x^4*(-3*x^4-2*x^2+1)^(1/2)+2*x^6*(-3*x^4-2*x^2+1)^(1/2)-2*x*(-3*x^4-2*x^2+1)^(1/2)-(-3*x^4-2*x^2+1)^(1/2)+x^2+3*x^4)/(3*x^6+2*x^6*(-3*x^4-2*x^2+1)^(1/2)+5*x^4+3*x^4*(-3*x^4-2*x^2+1)^(1/2)+x^2-1-(-3*x^4-2*x^2+1)^(1/2))) \\ _Joerg Arndt_, Nov 05 2013
%Y Cf. A111111, A230551-A230556.
%K nonn
%O 1,2
%A _Jay Pantone_, Nov 05 2013
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