Av(123, 132, 213, 3412)

Permutation examples

length 2: 12, 21

length 3: 231, 312, 321

length 4: 3421, 4231, 4312, 4321

length 5: 45321, 53421, 54231, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

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

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

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

Generating function

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

Coefficients

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

System of equations

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

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

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

$\operatorname{F_{1}}{\left (x \right )} = x \operatorname{F_{4}}{\left (x \right )}$

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