Av(123, 2413)

Permutation examples

length 2: 12, 21

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

length 4: 1432, 2143, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321

length 5: 15432, 21543, 25431, 31542, 32154, 32541, 35412, 35421, 41532, 42153, 42531, 43152, 43215, 43251, 43512, 43521, 45132, 45213, 45231, 45312, 45321, 51432, 52143, 52431, 53142, 53214, 53241, 53412, 53421, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 2413)

$\mathcal{B}$ = 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

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

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

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

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

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