A165538 -id:A165538 - cp's OEIS frontend

A278301 Number of permutations of length n in the class of juxtapositions of 321-avoiders with 21-avoiders.

1, 1, 2, 6, 23, 98, 434, 1949, 8803, 39888, 181201, 825201, 3767757, 17249560, 79191480, 364585230, 1683208515, 7792546040, 36174065285, 168367375735, 785637327745, 3674914227120, 17230132657815, 80965662243526, 381275131584373, 1799105397745998

Offset: 0

Jakub Sliacan, Nov 17 2016

nonn

a(n) is also the number of permutations of length n in the class of juxtapositions of 231-avoiders with 21-avoiders.

			There are 23 permutations of length 4 which can be expressed as a juxtaposition of a 321-avoider and a 21-avoider. Only 4321 is not expressable this way.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Robert Brignall, Jakub Sliacan, Juxtaposing Catalan permutation classes with monotone ones, arXiv:1611.05370 [math.CO], 2016.
Robert Brignall, Jakub Sliacan, Combinatorial specifications for juxtapositions of permutation classes, arXiv:1902.02705 [math.CO], 2019.

The other two juxtapositions of Catalan and monotone classes are enumerated by A033321, A165538.

e = ee /. Solve[ee == 1 + x/(1 - x) ee, ee][[1]];
c = cc /. Solve[cc == 1 + x cc^2, cc][[1]];
cb = ccb /. Solve[ccb == 1 + x/(1 - x) ccb^2, ccb][[2]];
b = bb /. Solve[bb == x^2/(1 - x) + x c bb e, bb][[1]];
m = mm /.
   Solve[mm ==
      x c mm cb + b e x/(1 - x) (cb - 1) + x^2/(1 - x) (cb - 1),
     mm][[1]];
f = c + c m cb/(1 - x);
CoefficientList[Series[f, {x, 0, 25}], x]
Rest[CoefficientList[Series[(1 - (1 - 4 x)^(1/2) + x (-4 + (1 - 4 x)^(1/2) + ((-1 + 5 x)^(1/2)) / ((-1 + x)^(1/2))))/ (-2 x^2), {x, 0, 33}], x]] (* Vincenzo Librandi, Nov 18 2016 *)

G.f.: (1 - (1-4*x)^(1/2) + x*(-4 + (1-4*x)^(1/2) + ((-1+5*x)^(1/2)) / ((-1+x)^(1/2)))) / (-2*x^2).

a(n) ~ 5^(n+3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 17 2016

A026876 T(2n+1, n+1), T given by A026758.

Original entry on oeis.org

1, 3, 11, 43, 174, 721, 3042, 13022, 56420, 246977, 1090846, 4856162, 21770665, 98217318, 445634370, 2032434475, 9313305870, 42861416044, 198038717055, 918368326483, 4273103732070, 19944292913477, 93356009312511, 438150836819153

Offset: 0

Clark Kimberling

nonn

The invert transform of this sequences (shifted) appears to be A165538 (conjectured). - Michael D. Weiner, Aug 04 2018

Cf. A026758, A165538.

G.f. f satisfies: x^6*f^4 + x^3*(4*x-1)*(x-1)*f^3 + x*(x-1)*(6*x^2-6*x+1)*f^2 + (4*x-1)*(x-1)^2*f + (x-1)^2 = 0 (conjectured). - Michael D. Weiner, Aug 04 2018

A326348 Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

Original entry on oeis.org

1, 1, 2, 6, 24, 115, 609, 3409, 19728, 116692, 701062, 4261581, 26146111, 161631115, 1005522262, 6289410686, 39525228204, 249427451071, 1579885391573, 10040587733693, 64004713573508, 409139527503760, 2622049900367018, 16843666877986873, 108438876033442579

Offset: 0

Robert Brignall, Sep 11 2019

nonn

			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.

Robert Brignall, Jakub Sliacan, Combinatorial specifications for juxtapositions of permutation classes, arXiv:1902.02705 [math.CO], 2019.

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

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 *)

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).

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

cp's OEIS Frontend

A278301 Number of permutations of length n in the class of juxtapositions of 321-avoiders with 21-avoiders.

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A026876 T(2n+1, n+1), T given by A026758.

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A326348 Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

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