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Av(123, 3412)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 231, 312, 321

length 4: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3421, 4132, 4213, 4231, 4312, 4321

length 5: 15432, 21543, 25143, 25413, 25431, 31542, 32154, 32514, 32541, 35214, 35241, 35421, 41532, 42153, 42531, 43152, 43215, 43251, 43521, 45321, 51432, 52143, 52413, 52431, 53142, 53214, 53241, 53421, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 11, depth 6)

Legend

$\mathcal{A}$ = Av(123, 3412)

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

Generating function

$A(x) = \frac{- 4 x^{4} + 9 x^{3} - 10 x^{2} + 5 x - 1}{2 x^{5} - 9 x^{4} + 16 x^{3} - 14 x^{2} + 6 x - 1}$

Coefficients

1, 1, 2, 5, 13, 33, 80, 185, 411, 885, 1862, ...

System of equations

$\operatorname{F_{25}}{\left (x \right )} = \operatorname{F_{15}}{\left (x \right )} + \operatorname{F_{24}}{\left (x \right )}$

$\operatorname{F_{15}}{\left (x \right )} = 1$

$\operatorname{F_{24}}{\left (x \right )} = \operatorname{F_{16}}{\left (x \right )} + \operatorname{F_{23}}{\left (x \right )}$

$\operatorname{F_{16}}{\left (x \right )} = - \frac{x}{x - 1}$

$\operatorname{F_{23}}{\left (x \right )} = \operatorname{F_{17}}{\left (x \right )} + \operatorname{F_{22}}{\left (x \right )}$

$\operatorname{F_{17}}{\left (x \right )} = x \left(\operatorname{F_{25}}{\left (x \right )} - 1\right)$

$\operatorname{F_{22}}{\left (x \right )} = \operatorname{F_{20}}{\left (x \right )} + \operatorname{F_{21}}{\left (x \right )}$

$\operatorname{F_{20}}{\left (x \right )} = \operatorname{F_{18}}{\left (x \right )} + \operatorname{F_{19}}{\left (x \right )}$

$\operatorname{F_{18}}{\left (x \right )} = - \frac{x^{3}}{\left(- 2 x + 1\right) \left(x - 1\right)}$

$\operatorname{F_{19}}{\left (x \right )} = - \frac{x^{4}}{\left(- 2 x + 1\right) \left(x - 1\right)^{3}}$

$\operatorname{F_{21}}{\left (x \right )} = - \frac{x^{3}}{\left(- 2 x + 1\right) \left(x - 1\right)}$