###### Av(132, 1234, 2314, 2341, 3124, 3241, 3421, 4213)
Generating Function
$$\displaystyle -\frac{\left(x +1\right) \left(x^{4}+x^{3}+2 x^{2}-x +1\right)}{x -1}$$
Counting Sequence
1, 1, 2, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(x -1\right) F \! \left(x \right)+\left(x +1\right) \left(x^{4}+x^{3}+2 x^{2}-x +1\right) = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 5$$
$$\displaystyle a \! \left(4\right) = 7$$
$$\displaystyle a \! \left(5\right) = 8$$
$$\displaystyle a \! \left(n \right) = 8, \quad n \geq 6$$
Explicit Closed Form
$$\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 5 & n =3 \\ 7 & n =4 \\ 8 & \text{otherwise} \end{array}\right.$$

### This specification was found using the strategy pack "Point Placements" and has 25 rules.

Found on January 18, 2022.

Finding the specification took 0 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)\\ \end{align*}