Av(132, 213, 231, 312)

Permutation examples

length 2: 12, 21

length 3: 123, 321

length 4: 1234, 4321

length 5: 12345, 54321

ATRAP tree (size 5, depth 3)

Legend

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

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

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

Generating function

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

Coefficients

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

System of equations

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

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

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

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

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