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Av(132, 3412)

Permutation examples

length 2: 12, 21

length 3: 123, 213, 231, 312, 321

length 4: 1234, 2134, 2314, 2341, 3124, 3214, 3241, 3421, 4123, 4213, 4231, 4312, 4321

length 5: 12345, 21345, 23145, 23415, 23451, 31245, 32145, 32415, 32451, 34215, 34251, 34521, 41235, 42135, 42315, 42351, 43125, 43215, 43251, 43521, 45321, 51234, 52134, 52314, 52341, 53124, 53214, 53241, 53421, 54123, 54213, 54231, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

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

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

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

Generating function

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

Coefficients

1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, ...

System of equations

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

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

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

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

$\operatorname{F_{2}}{\left (x \right )} = \frac{x^{2} \operatorname{F_{4}}{\left (x \right )}}{- 2 x + 1}$