Av(123, 132, 213)

Permutation examples

length 2: 12, 21

length 3: 231, 312, 321

length 4: 3412, 3421, 4231, 4312, 4321

length 5: 45231, 45312, 45321, 53412, 53421, 54231, 54312, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(123, 132, 213)

$\mathcal{B}$ = Av(12, 21)

Generating function

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

Coefficients

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

System of equations

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

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

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