ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 132, 321)

$\mathcal{B}$ = Av(21, 123)

$\mathcal{C}$ = Av(12, 321)

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

Generating function

$A(x) = x^{4} + 3 x^{3} + 2 x^{2} + x + 1$

Coefficients

1, 1, 2, 3, 1, 0, 0, 0, 0, 0, 0, ...

System of equations

$\operatorname{F_{9}}{\left (x \right )} = \operatorname{F_{5}}{\left (x \right )} + \operatorname{F_{8}}{\left (x \right )}$

$\operatorname{F_{5}}{\left (x \right )} = 1$

$\operatorname{F_{8}}{\left (x \right )} = \operatorname{F_{6}}{\left (x \right )} + \operatorname{F_{7}}{\left (x \right )}$

$\operatorname{F_{6}}{\left (x \right )} = x \left(x^{2} + x + 1\right)$

$\operatorname{F_{7}}{\left (x \right )} = x^{2} \left(x + 1\right)^{2}$