Av(123)

Permutation examples

length 2: 12, 21

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

length 4: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321

length 5: 15432, 21543, 25143, 25413, 25431, 31542, 32154, 32514, 32541, 35142, 35214, 35241, 35412, 35421, 41532, 42153, 42513, 42531, 43152, 43215, 43251, 43512, 43521, 45132, 45213, 45231, 45312, 45321, 51432, 52143, 52413, 52431, 53142, 53214, 53241, 53412, 53421, 54132, 54213, 54231, 54312, 54321

ATRAP tree (size 11, depth 5)

Note

The generating function for this class was calculated manually from the tree.

Legend

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

Generating function

$A(x) = \frac{1}{2 x} \left(- \sqrt{- 4 x + 1} + 1\right)$

Coefficients

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...

System of equations

$F{\left (x,y \right )} = x y \left(F{\left (x,y \right )} - 1\right) + x + \frac{x}{- y + 1} \left(- y \left(F{\left (x,y \right )} - 1\right) + F{\left (x,1 \right )} - 1\right) + 1$