ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 132, 213, 231, 312, 4321)

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

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

Generating function

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

Coefficients

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

System of equations

$\operatorname{F_{4}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{3}}{\left (x \right )}$

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

$\operatorname{F_{3}}{\left (x \right )} = \operatorname{F_{1}}{\left (x \right )} + \operatorname{F_{2}}{\left (x \right )}$

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

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