Av(2143, 2413, 3142, 3412)

Permutation examples

length 2: 12, 21

length 3: 123, 132, 213, 231, 312, 321

length 4: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2314, 2341, 2431, 3124, 3214, 3241, 3421, 4123, 4132, 4213, 4231, 4312, 4321

length 5: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13425, 13452, 13542, 14235, 14325, 14352, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 23145, 23415, 23451, 23541, 24315, 24351, 24531, 25341, 25431, 31245, 32145, 32415, 32451, 34215, 34251, 34521, 35421, 41235, 41325, 42135, 42315, 42351, 43125, 43215, 43251, 43521, 45321, 51234, 51243, 51324, 51342, 51423, 51432, 52134, 52314, 52341, 52431, 53124, 53214, 53241, 53421, 54123, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 9, depth 5)

Legend

$\mathcal{A}$ = Av(2143, 2413, 3142, 3412)

$\mathcal{B}$ = Av(12)

$\mathcal{C}$ = Av(21)

Generating function

$A(x) = \frac{- 3 x + 1}{2 x^{2} - 4 x + 1}$

Coefficients

1, 1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, ...

System of equations

$\operatorname{F_{30}}{\left (x \right )} = \operatorname{F_{22}}{\left (x \right )} + \operatorname{F_{29}}{\left (x \right )}$

$\operatorname{F_{22}}{\left (x \right )} = 1$

$\operatorname{F_{29}}{\left (x \right )} = \operatorname{F_{23}}{\left (x \right )} + \operatorname{F_{28}}{\left (x \right )}$

$\operatorname{F_{23}}{\left (x \right )} = x \operatorname{F_{30}}{\left (x \right )}$

$\operatorname{F_{28}}{\left (x \right )} = \operatorname{F_{26}}{\left (x \right )} + \operatorname{F_{27}}{\left (x \right )}$

$\operatorname{F_{26}}{\left (x \right )} = \operatorname{F_{24}}{\left (x \right )} + \operatorname{F_{25}}{\left (x \right )}$

$\operatorname{F_{24}}{\left (x \right )} = x \left(\operatorname{F_{30}}{\left (x \right )} - 1\right)$

$\operatorname{F_{25}}{\left (x \right )} = \frac{x^{2} \left(\operatorname{F_{30}}{\left (x \right )} - 1\right)}{- 2 x + 1}$

$\operatorname{F_{27}}{\left (x \right )} = \frac{x^{2} \left(\operatorname{F_{30}}{\left (x \right )} - 1\right)}{- 2 x + 1}$