Permutations avoiding 132, 2314, 3421, 4123, 4213, 4231

Proof tree for permutations avoiding 132, 2314, 3421, 4123, 4213, 4231

Legend

A = Av(021, 1203, 2310, 3012, 3102, 3120)

B = Av(021, 120, 201)

C = Av(021, 120, 3012, 3102)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}={\frac {x \left( {x}^{3}+2{x}^{2}+1 \right) }{ \left( x-1 \right) ^{2}}}$

Coefficients:
$1, 1, 2, 5, 9, 13, 17, 21, 25, 29,\ldots$

Minimal polynomial:
$ \left( x-1 \right) ^{2}F \left( x \right) -{x}^{4}-2{x}^{3}-{x}^{2}+x-1$

Generating function:
${\frac {{x}^{4}+2{x}^{3}+{x}^{2}-x+1}{ \left( x-1 \right) ^{2}}}$

Recurrence relation:
$a \left( n \right) -4n+7$
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$
$a \left( 4 \right) =9$

Closed form:
$\cases{1&$n=0$\cr 1&$n=1$\cr 2&$n=2$\cr -7+4n&otherwise\cr}$