### Av(1234, 1342, 1432, 2341, 2413, 3142, 3241, 4123, 4213)

#### Permutation examples

length 2: 12, 21

length 3: 123, 132, 213, 231, 312, 321

length 4: 1243, 1324, 1423, 2134, 2143, 2314, 2431, 3124, 3214, 3412, 3421, 4132, 4231, 4312, 4321

length 5: 13254, 21354, 21435, 21534, 23154, 24315, 31254, 32145, 32154, 34125, 34215, 35412, 35421, 41325, 42315, 43125, 43215, 45132, 45231, 45312, 45321, 52431, 53412, 53421, 54132, 54231, 54312, 54321

#### Conjectured Struct Cover

 $\mathcal{A}$
$=$
 $\bullet$ $\bullet$ $\bullet$ \ $\bullet$ $\bullet$
$\bigsqcup$
 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\mathcal{B}$
$\bigsqcup$
 $\bullet$ $\bullet$ \ $\bullet$ $\bullet$
$\bigsqcup$
 $\bullet$ $\bullet$ $\bullet$ \ $\bullet$ $\bullet$
$\bigsqcup$
 $\bullet$ $\bullet$ $\bullet$ $\bullet$
$\bigsqcup$
 $\mathcal{B}$ $\bullet$ $\bullet$ $\bullet$ $\mathcal{C}$
$\bigsqcup$
 $\bullet$ $\mathcal{B}$ $\bullet$ $\bullet$ $\bullet$ $\mathcal{C}$
$\bigsqcup$
 $\bullet$ $\bullet$ $\bullet$ $\bullet$ \ $\bullet$ $\bullet$
$\bigsqcup$
 $\bullet$ $\bullet$ \ $\bullet$ $\bullet$ $\bullet$ $\mathcal{B}$
$\bigsqcup$
 $\mathcal{D}$

#### Recurrence relation

$a_{0} = 1,~a_{1} = 1,~a_{2} = 2,~a_{3} = 6,~a_{4} = 15,~a_{5} = 28,~a_{6} = 48$

$a_{n} = b_{n-4}+\sum_{i=0}^{n-3}b_{i}c_{n-i-3}+\sum_{i=0}^{n-4}b_{i}c_{n-i-4}+\sum_{i=0}^{n-5}2b_{n-i-5}+d_{n}+5$