Av(132, 213, 231, 4123, 4321)

Permutation examples

length 2: 12, 21

length 3: 123, 312, 321

length 4: 1234, 4312

length 5: 12345

ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(132, 213, 231, 4123, 4321)

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

$\mathcal{C}$ = Av(123, 132, 213, 231, 321)

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

Generating function

$A(x) = \frac{1}{x - 1} \left(x^{5} + x^{4} - x^{3} - x^{2} - 1\right)$

Coefficients

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

System of equations

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

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

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

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

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