Av(132, 213)

Permutation examples

length 2: 12, 21

length 3: 123, 231, 312, 321

length 4: 1234, 2341, 3412, 3421, 4123, 4231, 4312, 4321

length 5: 12345, 23451, 34512, 34521, 45123, 45231, 45312, 45321, 51234, 52341, 53412, 53421, 54123, 54231, 54312, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(132, 213)

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

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

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

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

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