Permutations avoiding 132, 934256781

Proof tree for permutations avoiding 132, 934256781

Legend

A = Av(120, 076542138)

B = Av(120, 07654213)

C = Av(120, 65431027)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}={ F_{12}}{ F_{8}}$
${ F_{12}}={ F_{0}}+{ F_{14}}$
${ F_{14}}={ F_{20}}{ F_{21}}$
${ F_{20}}={\frac {-4{x}^{3}+10{x}^{2}-6x+1}{{x}^{4}-10{x}^{3}+15{x}^{2}-7x+1}}$
${ F_{21}}=-{\frac {x \left( {x}^{3}-6{x}^{2}+5x-1 \right) }{{x}^{4}-10{x}^{3}+15{x}^{2}-7x+1}}$
${ F_{8}}=x$

Coefficients:
$1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861,\ldots$

Minimal polynomial:
$ \left( {x}^{3}-9{x}^{2}+6x-1 \right) ^{2} \left( x-1 \right) ^{3}F \left( x \right) +5{x}^{8}-54{x}^{7}+216{x}^{6}-405{x}^{5}+413{x}^{4}-241{x}^{3}+80{x}^{2}-14x+1$

Generating function:
$-{\frac {5{x}^{8}-54{x}^{7}+216{x}^{6}-405{x}^{5}+413{x}^{4}-241{x}^{3}+80{x}^{2}-14x+1}{ \left( {x}^{3}-9{x}^{2}+6x-1 \right) ^{2} \left( x-1 \right) ^{3}}}$

Recurrence relation:
$-2a \left( n \right) +36a \left( n+1 \right) -186a \left( n+2 \right) +220a \left( n+3 \right) -108a \left( n+4 \right) +24a \left( n+5 \right) -2a \left( n+6 \right) +{n}^{2}-n+6$
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$
$a \left( 4 \right) =14$
$a \left( 5 \right) =42$
$a \left( 6 \right) =132$
$a \left( 7 \right) =429$
$a \left( 8 \right) =1430$

Closed form:
${\frac { \left( {\frac { \left( -18i\sqrt {3}-20 \right) \sqrt [3]{2} \left( 37+i\sqrt {3} \right) ^{2/3}}{196}}+i/4\sqrt {3}\sqrt [3]{148+4i\sqrt {3}}-1/4\sqrt [3]{148+4i\sqrt {3}}+3 \right) ^{-n}}{15876} \left( -119 \left( \left( in-{\frac {15i}{17}} \right) \sqrt {3}-{\frac {13n}{17}}+{\frac{23}{17}} \right) {2}^{2/3}\sqrt [3]{37+i\sqrt {3}}+44\sqrt [3]{2} \left( \left( in-{\frac {19i}{22}} \right) \sqrt {3}+{\frac {19n}{22}}-{\frac{16}{11}} \right) \left( 37+i\sqrt {3} \right) ^{2/3}-980n+2352 \right) }+{\frac { \left( {\frac { \left( -49i\sqrt {3}-49 \right) \sqrt [3]{148+4i\sqrt {3}}}{196}}+3+{\frac { \left( 19i\sqrt {3}-17 \right) \sqrt [3]{2} \left( 37+i\sqrt {3} \right) ^{2/3}}{196}} \right) ^{-n}}{15876} \left( 105 \left( \left( in-{\frac {19i}{15}} \right) \sqrt {3}+{\frac {19n}{15}}-{\frac{11}{15}} \right) {2}^{2/3}\sqrt [3]{37+i\sqrt {3}}-41\sqrt [3]{2} \left( \left( in-{\frac {51i}{41}} \right) \sqrt {3}-{\frac {47n}{41}}+{\frac{25}{41}} \right) \left( 37+i\sqrt {3} \right) ^{2/3}-980n+2352 \right) }+{\frac { \left( {\frac { \left( -i\sqrt {3}+37 \right) \sqrt [3]{2} \left( 37+i\sqrt {3} \right) ^{2/3}}{196}}+1/2\sqrt [3]{148+4i\sqrt {3}}+3 \right) ^{-n}}{15876} \left( 14 \left( \left( in+2i \right) \sqrt {3}-16n+17 \right) {2}^{2/3}\sqrt [3]{37+i\sqrt {3}}-3\sqrt [3]{2} \left( \left( in+13/3i \right) \sqrt {3}+{\frac {85n}{3}}-{\frac{89}{3}} \right) \left( 37+i\sqrt {3} \right) ^{2/3}-980n+2352 \right) }+1/18{n}^{2}-n/18+5/9$