Permutations avoiding 132, 213, 1234, 2341, 3421, 4123, 4321

Proof tree for permutations avoiding 132, 213, 1234, 2341, 3421, 4123, 4321

Legend

A = Av(021, 102, 0123, 1230, 2310, 3012, 3210)

B = Av(012, 021, 102, 120, 210)

C = Av(10, 012)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}=x \left( {x}^{4}+3{x}^{3}+4{x}^{2}+2x+1 \right) $

Coefficients:
$1, 1, 2, 4, 3, 1, 0, 0, 0, 0,\ldots$

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

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

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

Closed form:
$\cases{1&$n=0$\cr 1&$n=1$\cr 2&$n=2$\cr 4&$n=3$\cr 3&$n=4$\cr 1&$n=5$\cr 0&otherwise\cr}$