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

0,3

Comments

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

Links

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.

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A150298 A280768 A370183 * A242586 A196018 A009449

Adjacent sequences: A278298 A278299 A278300 * A278302 A278303 A278304

Keyword

nonn

Author

Jakub Sliacan, Nov 17 2016

Status

approved