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Av(123, 231, 1432)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 312, 321

length 4: 2143, 3214, 4132, 4213, 4312, 4321

length 5: 32154, 43215, 52143, 53214, 54132, 54213, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 231, 1432)

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

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

Generating function

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

Coefficients

1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

System of equations

$\operatorname{F_{7}}{\left (x \right )} = \operatorname{F_{3}}{\left (x \right )} + \operatorname{F_{6}}{\left (x \right )}$

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

$\operatorname{F_{6}}{\left (x \right )} = \operatorname{F_{4}}{\left (x \right )} + \operatorname{F_{5}}{\left (x \right )}$

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

$\operatorname{F_{5}}{\left (x \right )} = \frac{x^{2} \left(x + 1\right)}{\left(x - 1\right)^{2}}$