OFFSET
1,2
COMMENTS
The separable permutations are those avoiding 2413 and 3142 and are counted by the large Schroeder numbers (A006318).
The involutions avoiding 2413 coincide with the involutions avoiding 3142, and hence both sets coincide with the separable involutions. - David Callan, Aug 27 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..600
Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013-2014.
R. Brignall, S. Huczynska and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.
FORMULA
G.f. f satisfies: x^2f^4 + (x^3+3x^2+x-1)f^3 + (3x^3+6x^2-x)f^2 + (3x^3+7x^2-x-1)f +x^3+3x^2+x=0.
a(n) ~ sqrt(6 + 6*sqrt(2) + 4*sqrt(3) + 3*sqrt(6)) * (5+2*sqrt(6))^(n/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 13 2014
EXAMPLE
a(5) = 24 because of the 26 involutions of length 5 only two are not separable, 35142 and 42513.
MATHEMATICA
terms = 30;
f[_] = 0; Do[f[x_] = Normal[(-(x^3 f[x]^3) - 3 x^3 f[x]^2 - x^2 f[x]^4 - 3 x^2 f[x]^3 - 6 x^2 f[x]^2 - x f[x]^3 + f[x]^3 + x f[x]^2 - x^3 - 3 x^2 - x)/(3 x^3 + 7 x^2 - x - 1) + O[x]^(terms+1)], {terms+1}];
CoefficientList[f[x]/x, x] (* Jean-François Alcover, Nov 05 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vincent Vatter, Aug 16 2006
STATUS
approved