Av(1342, 1432)

Permutation examples

length 2: 12, 21

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

length 4: 1234, 1243, 1324, 1423, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321

length 5: 12345, 12354, 12435, 12534, 13245, 13254, 14235, 15234, 21345, 21354, 21435, 21534, 23145, 23154, 23415, 23451, 23514, 23541, 24135, 24315, 24351, 25134, 25314, 25341, 31245, 31254, 31425, 31524, 32145, 32154, 32415, 32451, 32514, 32541, 34125, 34152, 34215, 34251, 34512, 34521, 35124, 35142, 35214, 35241, 35412, 35421, 41235, 41253, 41325, 41523, 42135, 42153, 42315, 42351, 42513, 42531, 43125, 43152, 43215, 43251, 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 51234, 51243, 51324, 51423, 52134, 52143, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241, 53412, 53421, 54123, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 27, depth 12)

Legend

$\mathcal{A}$ = Av(1342, 1432)

$\mathcal{B}$ = Av(231, 321)

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

Generating function

$A(x) = - \frac{x}{2} - \frac{1}{2} \sqrt{x^{2} - 6 x + 1} + \frac{3}{2}$

Coefficients

1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, ...

System of equations

$\operatorname{F_{6}}{\left (x \right )} = \operatorname{F_{18}}{\left (x \right )} + \operatorname{F_{363}}{\left (x \right )}$

$\operatorname{F_{363}}{\left (x \right )} = 1$

$\operatorname{F_{18}}{\left (x \right )} = \operatorname{F_{364}}{\left (x \right )} + \operatorname{F_{372}}{\left (x \right )}$

$\operatorname{F_{364}}{\left (x \right )} = \frac{x \left(x - 1\right)}{2 x - 1}$

$\operatorname{F_{372}}{\left (x \right )} = \operatorname{F_{366}}{\left (x \right )} + \operatorname{F_{371}}{\left (x \right )}$

$\operatorname{F_{366}}{\left (x \right )} = \operatorname{F_{138}}{\left (x \right )} \operatorname{F_{365}}{\left (x \right )}$

$\operatorname{F_{365}}{\left (x \right )} = x \left(\operatorname{F_{6}}{\left (x \right )} - 1\right)$

$\operatorname{F_{138}}{\left (x \right )} = \frac{x - 1}{2 x - 1}$

$\operatorname{F_{371}}{\left (x \right )} = \operatorname{F_{230}}{\left (x \right )} \operatorname{F_{30}}{\left (x \right )}$

$\operatorname{F_{30}}{\left (x \right )} = \operatorname{F_{368}}{\left (x \right )} + \operatorname{F_{370}}{\left (x \right )}$

$\operatorname{F_{368}}{\left (x \right )} = \operatorname{F_{18}}{\left (x \right )} \operatorname{F_{5}}{\left (x \right )}$

$\operatorname{F_{5}}{\left (x \right )} = \operatorname{F_{365}}{\left (x \right )} + \operatorname{F_{76}}{\left (x \right )}$

$\operatorname{F_{76}}{\left (x \right )} = \operatorname{F_{370}}{\left (x \right )} + \operatorname{F_{87}}{\left (x \right )}$

$\operatorname{F_{87}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} \operatorname{F_{52}}{\left (x \right )}$

$\operatorname{F_{52}}{\left (x \right )} = \operatorname{F_{369}}{\left (x \right )} + \operatorname{F_{76}}{\left (x \right )}$

$\operatorname{F_{369}}{\left (x \right )} = \operatorname{F_{367}}{\left (x \right )} + \operatorname{F_{368}}{\left (x \right )}$

$\operatorname{F_{367}}{\left (x \right )} = x \left(\operatorname{F_{6}}{\left (x \right )} - 1\right)$

$\operatorname{F_{0}}{\left (x \right )} = x$

$\operatorname{F_{370}}{\left (x \right )} = \operatorname{F_{5}}{\left (x \right )} \operatorname{F_{52}}{\left (x \right )}$

$\operatorname{F_{230}}{\left (x \right )} = - \frac{1}{2 x - 1}$