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

1,2

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

Cf. A121703.

Sequence in context: A132220 A007874 A294410 * A049144 A354733 A336419

Adjacent sequences: A121701 A121702 A121703 * A121705 A121706 A121707

Keyword

nonn

Author

Vincent Vatter, Aug 16 2006

Status

approved