Av(132, 4231)

Permutation examples

length 2: 12, 21

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

length 4: 1234, 2134, 2314, 2341, 3124, 3214, 3241, 3412, 3421, 4123, 4213, 4312, 4321

length 5: 12345, 21345, 23145, 23415, 23451, 31245, 32145, 32415, 32451, 34125, 34215, 34251, 34512, 34521, 41235, 42135, 43125, 43215, 43251, 43512, 43521, 45123, 45213, 45312, 45321, 51234, 52134, 53124, 53214, 54123, 54213, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(132, 4231)

$\mathcal{B}$ = Av(132, 312)

$\mathcal{C}$ = Av(132, 231)

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

Generating function

$A(x) = \frac{x^{3} - 5 x^{2} + 4 x - 1}{4 x^{3} - 8 x^{2} + 5 x - 1}$

Coefficients

1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, ...

System of equations

$\operatorname{F_{9}}{\left (x \right )} = \operatorname{F_{5}}{\left (x \right )} + \operatorname{F_{8}}{\left (x \right )}$

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

$\operatorname{F_{8}}{\left (x \right )} = \operatorname{F_{6}}{\left (x \right )} + \operatorname{F_{7}}{\left (x \right )}$

$\operatorname{F_{6}}{\left (x \right )} = x \operatorname{F_{9}}{\left (x \right )}$

$\operatorname{F_{7}}{\left (x \right )} = - \frac{x^{2} \left(x - 1\right)}{\left(2 x - 1\right)^{2}}$