Av(1342, 2431)

Permutation examples

length 2: 12, 21

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

length 4: 1234, 1243, 1324, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321

length 5: 12345, 12354, 12435, 12534, 12543, 13245, 13254, 14235, 14325, 15234, 15243, 15324, 15423, 15432, 21345, 21354, 21435, 21534, 21543, 23145, 23154, 23415, 23451, 23514, 24135, 25134, 25143, 31245, 31254, 31425, 31524, 32145, 32154, 32415, 32451, 32514, 34125, 34152, 34215, 34251, 34512, 34521, 35124, 35214, 41235, 41253, 41325, 41523, 41532, 42135, 42153, 42315, 42351, 42513, 43125, 43152, 43215, 43251, 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 51234, 51243, 51324, 51423, 51432, 52134, 52143, 52314, 52341, 52413, 53124, 53142, 53214, 53241, 53412, 53421, 54123, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 14, depth 6)

Legend

$\mathcal{A}$ = Av(1342, 2431)

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

$\mathcal{C}$ = Av(132)

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

Generating function

$A(x) = \frac{- x^{3} \sqrt{- 4 x + 1} - 9 x^{3} + 8 x^{2} \sqrt{- 4 x + 1} + 18 x^{2} - 6 x \sqrt{- 4 x + 1} - 8 x + \sqrt{- 4 x + 1} + 1}{4 x^{4} - 6 x^{3} \sqrt{- 4 x + 1} - 22 x^{3} + 13 x^{2} \sqrt{- 4 x + 1} + 25 x^{2} - 7 x \sqrt{- 4 x + 1} - 9 x + \sqrt{- 4 x + 1} + 1}$

Coefficients

1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, ...

System of equations

$\operatorname{F_{49}}{\left (x \right )} = \operatorname{F_{60}}{\left (x \right )} + \operatorname{F_{8}}{\left (x \right )}$

$\operatorname{F_{60}}{\left (x \right )} = 1$

$\operatorname{F_{8}}{\left (x \right )} = \operatorname{F_{58}}{\left (x \right )} + \operatorname{F_{61}}{\left (x \right )}$

$\operatorname{F_{61}}{\left (x \right )} = x \operatorname{F_{49}}{\left (x \right )}$

$\operatorname{F_{58}}{\left (x \right )} = \operatorname{F_{64}}{\left (x \right )} + \operatorname{F_{65}}{\left (x \right )}$

$\operatorname{F_{64}}{\left (x \right )} = \operatorname{F_{62}}{\left (x \right )} + \operatorname{F_{63}}{\left (x \right )}$

$\operatorname{F_{62}}{\left (x \right )} = x \left(\operatorname{F_{49}}{\left (x \right )} - 1\right)$

$\operatorname{F_{63}}{\left (x \right )} = \frac{1}{2} \left(- \sqrt{- 4 x + 1} + 1\right) \left(- \frac{1}{2} \sqrt{- 4 x + 1} + \frac{1}{2}\right) \left(\operatorname{F_{49}}{\left (x \right )} - 1\right)$

$\operatorname{F_{65}}{\left (x \right )} = \operatorname{F_{19}}{\left (x \right )} \operatorname{F_{6}}{\left (x \right )}$

$\operatorname{F_{19}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} \operatorname{F_{3}}{\left (x \right )} \operatorname{F_{58}}{\left (x \right )}$

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

$\operatorname{F_{3}}{\left (x \right )} = \frac{1}{2 x} \left(- \sqrt{- 4 x + 1} + 1\right)$

$\operatorname{F_{6}}{\left (x \right )} = \frac{1}{2 x} \left(- \sqrt{- 4 x + 1} + 1\right)$