Permutations avoiding 132, 1234, 2134, 2341, 3241, 3412, 4213, 4312, 4321

Proof tree for permutations avoiding 132, 1234, 2134, 2341, 3241, 3412, 4213, 4312, 4321

Legend

A = Av(021, 0123, 1023, 1230, 2130, 2301, 3102, 3201, 3210)

B = Av(012, 021, 102, 210, 2301)

C = Av(012, 021, 102, 2301, 3201, 3210)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}=2{x}^{5}+6{x}^{4}+5{x}^{3}+2{x}^{2}+x$

Coefficients:
$1, 1, 2, 5, 6, 2, 0, 0, 0, 0,\ldots$

Minimal polynomial:
$-2{x}^{5}-6{x}^{4}-5{x}^{3}-2{x}^{2}+F \left( x \right) -x-1$

Generating function:
$2{x}^{5}+6{x}^{4}+5{x}^{3}+2{x}^{2}+x+1$

Recurrence relation:
$a \left( n \right) $
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$
$a \left( 4 \right) =6$
$a \left( 5 \right) =2$

Closed form:
$\cases{1&$n=0$\cr 1&$n=1$\cr 2&$n=2$\cr 5&$n=3$\cr 6&$n=4$\cr 2&$n=5$\cr 0&otherwise\cr}$