###### Av(132, 1234, 2134, 2314, 3124, 3412)
Generating Function
$$\displaystyle \frac{x^{4}-x^{3}-2 x^{2}+2 x -1}{\left(x -1\right)^{3}}$$
Counting Sequence
1, 1, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, ...
Implicit Equation for the Generating Function
$$\displaystyle -\left(x -1\right)^{3} F \! \left(x \right)+x^{4}-x^{3}-2 x^{2}+2 x -1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 5$$
$$\displaystyle a \! \left(4\right) = 9$$
$$\displaystyle a \! \left(n \right) = \frac{\left(n +2\right) \left(-1+n \right)}{2}, \quad n \geq 5$$
Explicit Closed Form
$$\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n +2\right) \left(-1+n \right)}{2} & \text{otherwise} \end{array}\right.$$

### 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_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)\\ \end{align*}