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Av(132, 213, 2341)

Permutation examples

length 2: 12, 21

length 3: 123, 231, 312, 321

length 4: 1234, 3412, 3421, 4123, 4231, 4312, 4321

length 5: 12345, 45123, 45231, 45312, 45321, 51234, 53412, 53421, 54123, 54231, 54312, 54321

ATRAP tree (size 3, depth 2)

Legend

$\mathcal{A}$ = Av(132, 213, 2341)

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

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

Generating function

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

Coefficients

1, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, ...

System of equations

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

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

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