Av(123, 132, 231)

Permutation examples

length 2: 12, 21

length 3: 213, 312, 321

length 4: 3214, 4213, 4312, 4321

length 5: 43215, 53214, 54213, 54312, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(123, 132, 231)

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

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

Generating function

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

Coefficients

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

System of equations

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

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

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