ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 132, 213, 3412, 3421)

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

$\mathcal{C}$ = Av(123, 132, 213, 231)

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

Generating function

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

Coefficients

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

System of equations

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

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

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

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

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