Av(123, 132, 3412)

Permutation examples

length 2: 12, 21

length 3: 213, 231, 312, 321

length 4: 3214, 3241, 3421, 4213, 4231, 4312, 4321

length 5: 43215, 43251, 43521, 45321, 53214, 53241, 53421, 54213, 54231, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

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

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

Generating function

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

Coefficients

1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...

System of equations

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

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

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

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

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