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Av(123, 132)

Permutation examples

length 2: 12, 21

length 3: 213, 231, 312, 321

length 4: 3214, 3241, 3412, 3421, 4213, 4231, 4312, 4321

length 5: 43215, 43251, 43512, 43521, 45213, 45231, 45312, 45321, 53214, 53241, 53412, 53421, 54213, 54231, 54312, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(123, 132)

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

Generating function

$A(x) = \frac{x - 1}{2 x - 1}$

Coefficients

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...

System of equations

$\operatorname{F_{2}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{1}}{\left (x \right )}$

$\operatorname{F_{0}}{\left (x \right )} = 1$

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