A326348 - OEIS

%I #22 Jul 07 2024 10:29:27

%S 1,1,2,6,24,115,609,3409,19728,116692,701062,4261581,26146111,

%T 161631115,1005522262,6289410686,39525228204,249427451071,

%U 1579885391573,10040587733693,64004713573508,409139527503760,2622049900367018,16843666877986873,108438876033442579

%N Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

%H Robert Brignall, Jakub Sliacan, <a href="https://arxiv.org/abs/1902.02705">Combinatorial specifications for juxtapositions of permutation classes</a>, arXiv:1902.02705 [math.CO], 2019.

%F G.f.: (2-4*z+z^2)*x*y/(4*(1-z)*(-2+7*z-7*z^2+z^3)) + ((-2+10*z-15*z^2+7*z^3)*x + (2-6*z+z^2+6*z^3-z^4)*y - 10+54*z-99*z^2+66*z^3-9z^4)/(4*(1-z)^2*(-2+7*z-7*z^2+z^3)) where x=sqrt(1-6*z+z^2) and y=sqrt(1-8*z+8z^2).

%F a(n) ~ (63 + 8*sqrt(2) + 3*sqrt(41 + 40*sqrt(2))) * 2^(3*n/2 - 1) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * (73 + 53*sqrt(2)) * n^(3/2)). - _Vaclav Kotesovec_, Jul 07 2024

%e There are a(5) = 115 permutations of length 5 which can be expressed as a juxtaposition of a separable permutation (avoiding 2413 and 3142) with an increasing permutation. These 5 cannot be expressed: 25143, 35142, 35241, 41532 and 42531.

%t CoefficientList[Series[(Sqrt[1 - 6*x + x^2]*(2 - 4*x + x^2)*Sqrt[1 - 8*x + 8*x^2]) / (4*(1 - x)*(-2 + 7*x - 7*x^2 + x^3)) + (-10 + 54*x - 99*x^2 + 66*x^3 - 9*x^4 + Sqrt[1 - 6*x + x^2]*(-2 + 10*x - 15*x^2 + 7*x^3) + Sqrt[1 - 8*x + 8*x^2]*(2 - 6*x + x^2 + 6*x^3 - x^4))/(4*(1 - x)^2*(-2 + 7*x - 7*x^2 + x^3)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Jul 07 2024 *)

%Y Other juxtapositions of algebraic classes with monotone ones are enumerated by A033321, A165538, and A278301.

%K nonn

%O 0,3

%A _Robert Brignall_, Sep 11 2019