###### Av(1234, 1243, 1324, 1342, 1423, 2314, 3412)
Generating Function
$$\displaystyle -\frac{2 x^{3}-6 x^{2}+4 x -1}{\left(x -1\right)^{3} \left(2 x -1\right)}$$
Counting Sequence
1, 1, 2, 6, 17, 43, 100, 220, 467, 969, 1982, 4018, 8101, 16279, 32648, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(x -1\right)^{3} \left(2 x -1\right) F \! \left(x \right)+2 x^{3}-6 x^{2}+4 x -1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n +2\right) \left(n -1\right)}{2}, \quad n \geq 4$$
Explicit Closed Form
$$\displaystyle 2^{n +1}-\frac{n^{2}}{2}-\frac{3 n}{2}-1$$

### This specification was found using the strategy pack "Point Placements" and has 35 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\ \end{align*}