A121704 Number of separable involutions.

1, 2, 4, 10, 24, 64, 166, 456, 1234, 3454, 9600, 27246, 77132, 221336, 635078, 1839000, 5331274, 15555586, 45465412, 133517130, 392841336, 1160033656, 3432015726, 10182891552, 30267591290, 90177226062, 269117947728, 804699330974, 2409839825756, 7228746487536

Offset: 1

Author

Vincent Vatter, Aug 16 2006

Keywords

• nonn

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

Examples

			a(5) = 24 because of the 26 involutions of length 5 only two are not separable, 35142 and 42513.
		

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.

Crossrefs

Cf. A121703.

Programs

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 *)

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