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Av(123, 231, 2143)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 312, 321

length 4: 1432, 3214, 4132, 4213, 4312, 4321

length 5: 15432, 43215, 51432, 53214, 54132, 54213, 54312, 54321

ATRAP tree (size 7, depth 4)

Legend

$\mathcal{A}$ = Av(123, 231, 2143)

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

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

$\operatorname{F_{6}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{5}}{\left (x \right )}$

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

$\operatorname{F_{5}}{\left (x \right )} = \operatorname{F_{1}}{\left (x \right )} + \operatorname{F_{4}}{\left (x \right )}$

$\operatorname{F_{1}}{\left (x \right )} = x \operatorname{F_{6}}{\left (x \right )}$

$\operatorname{F_{4}}{\left (x \right )} = \operatorname{F_{2}}{\left (x \right )} + \operatorname{F_{3}}{\left (x \right )}$

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

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