Av(1324, 2143)

Permutation examples

length 2: 12, 21

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

length 4: 1234, 1243, 1342, 1423, 1432, 2134, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321

length 5: 12345, 12354, 12453, 12534, 12543, 13452, 13542, 14523, 14532, 15234, 15243, 15342, 15423, 15432, 21345, 23145, 23415, 23451, 23514, 23541, 24513, 24531, 25134, 25314, 25341, 25413, 25431, 31245, 31452, 32145, 32415, 32451, 34125, 34152, 34215, 34251, 34512, 34521, 35124, 35142, 35214, 35241, 35412, 35421, 41235, 41253, 41352, 41523, 41532, 42135, 42315, 42351, 42513, 42531, 43125, 43152, 43215, 43251, 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 51234, 51243, 51342, 51423, 51432, 52134, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241, 53412, 53421, 54123, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 16, depth 7)

Legend

$\mathcal{A}$ = Av(1324, 2143)

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

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

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

Generating function

$A(x) = \frac{x - \sqrt{- 4 x + 1}}{x \sqrt{- 4 x + 1} + x - \sqrt{- 4 x + 1}}$

Coefficients

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

System of equations

$\operatorname{F_{82}}{\left (x \right )} = \operatorname{F_{180}}{\left (x \right )} + \operatorname{F_{67}}{\left (x \right )}$

$\operatorname{F_{180}}{\left (x \right )} = 1$

$\operatorname{F_{67}}{\left (x \right )} = \operatorname{F_{137}}{\left (x \right )} + \operatorname{F_{181}}{\left (x \right )}$

$\operatorname{F_{181}}{\left (x \right )} = x \operatorname{F_{82}}{\left (x \right )}$

$\operatorname{F_{137}}{\left (x \right )} = \operatorname{F_{184}}{\left (x \right )} + \operatorname{F_{185}}{\left (x \right )}$

$\operatorname{F_{184}}{\left (x \right )} = \operatorname{F_{182}}{\left (x \right )} + \operatorname{F_{183}}{\left (x \right )}$

$\operatorname{F_{182}}{\left (x \right )} = - \frac{x^{2} \operatorname{F_{82}}{\left (x \right )}}{x - 1}$

$\operatorname{F_{183}}{\left (x \right )} = \operatorname{F_{179}}{\left (x \right )} \operatorname{F_{21}}{\left (x \right )}$

$\operatorname{F_{179}}{\left (x \right )} = \operatorname{F_{1}}{\left (x \right )} \operatorname{F_{137}}{\left (x \right )} \operatorname{F_{7}}{\left (x \right )}$

$\operatorname{F_{7}}{\left (x \right )} = x$

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

$\operatorname{F_{21}}{\left (x \right )} = - \frac{1}{x - 1}$

$\operatorname{F_{185}}{\left (x \right )} = \operatorname{F_{137}}{\left (x \right )} \operatorname{F_{14}}{\left (x \right )}$

$\operatorname{F_{14}}{\left (x \right )} = - \frac{1}{x - 1} \left(- \frac{1}{2} \sqrt{- 4 x + 1} + \frac{1}{2}\right)$