Av(1342, 3241)

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, 2431, 3124, 3142, 3214, 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, 23541, 24135, 24315, 25134, 25143, 25314, 25341, 25413, 25431, 31245, 31254, 31425, 31524, 31542, 32145, 32154, 34125, 34152, 34215, 34512, 34521, 35124, 35142, 35214, 35412, 35421, 41235, 41253, 41325, 41523, 41532, 42135, 42153, 42315, 43125, 43215, 45123, 45132, 45213, 45231, 45312, 45321, 51234, 51243, 51324, 51423, 51432, 52134, 52143, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53412, 53421, 54123, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 26, depth 10)

Legend

$\mathcal{A}$ = Av(1342, 3241)

$\mathcal{B}$ = Av(213, 1342)

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

$\mathcal{D}$ = Av(213, 231)

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

Coefficients

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

System of equations

$\operatorname{F_{257}}{\left (x \right )} = \operatorname{F_{2003}}{\left (x \right )} + \operatorname{F_{37}}{\left (x \right )}$

$\operatorname{F_{2003}}{\left (x \right )} = 1$

$\operatorname{F_{37}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{2012}}{\left (x \right )}$

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

$\operatorname{F_{2012}}{\left (x \right )} = \operatorname{F_{11}}{\left (x \right )} + \operatorname{F_{367}}{\left (x \right )}$

$\operatorname{F_{11}}{\left (x \right )} = \operatorname{F_{1545}}{\left (x \right )} + \operatorname{F_{2004}}{\left (x \right )}$

$\operatorname{F_{2004}}{\left (x \right )} = \frac{1}{2} \left(- \sqrt{- 4 x + 1} + 1\right) \left(- \frac{1}{2} \sqrt{- 4 x + 1} + \frac{1}{2}\right)$

$\operatorname{F_{1545}}{\left (x \right )} = \operatorname{F_{2009}}{\left (x \right )} + \operatorname{F_{2011}}{\left (x \right )} + \operatorname{F_{378}}{\left (x \right )}$

$\operatorname{F_{378}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} \operatorname{F_{315}}{\left (x \right )}$

$\operatorname{F_{315}}{\left (x \right )} = \operatorname{F_{2005}}{\left (x \right )} + \operatorname{F_{2007}}{\left (x \right )}$

$\operatorname{F_{2005}}{\left (x \right )} = x \left(\operatorname{F_{257}}{\left (x \right )} - 1\right)$

$\operatorname{F_{2007}}{\left (x \right )} = \operatorname{F_{1545}}{\left (x \right )} + \operatorname{F_{2006}}{\left (x \right )}$

$\operatorname{F_{2006}}{\left (x \right )} = \operatorname{F_{315}}{\left (x \right )} \operatorname{F_{97}}{\left (x \right )}$

$\operatorname{F_{97}}{\left (x \right )} = - \frac{x}{x - 1}$

$\operatorname{F_{2009}}{\left (x \right )} = \operatorname{F_{2008}}{\left (x \right )} \operatorname{F_{905}}{\left (x \right )}$

$\operatorname{F_{2008}}{\left (x \right )} = \frac{x^{3}}{\sqrt{- 4 x + 1}}$

$\operatorname{F_{905}}{\left (x \right )} = - \frac{1}{2 x \left(x - 1\right)} \left(- \sqrt{- 4 x + 1} + 1\right) \left(\operatorname{F_{257}}{\left (x \right )} - 1\right)$

$\operatorname{F_{2011}}{\left (x \right )} = \operatorname{F_{1545}}{\left (x \right )} \operatorname{F_{2010}}{\left (x \right )}$

$\operatorname{F_{2010}}{\left (x \right )} = - \frac{1}{2} \sqrt{- 4 x + 1} + \frac{1}{2}$

$\operatorname{F_{367}}{\left (x \right )} = \operatorname{F_{2005}}{\left (x \right )} + \operatorname{F_{2006}}{\left (x \right )}$