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Av(123, 231, 312)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 321

length 4: 1432, 2143, 3214, 4321

length 5: 15432, 21543, 32154, 43215, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(123, 231, 312)

$\mathcal{B}$ = 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

$\operatorname{F_{2}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{1}}{\left (x \right )}$

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

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