###### Av(123, 3412)
Generating Function
$$\displaystyle -\frac{4 x^{4}-9 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}$$
Counting Sequence
1, 1, 2, 5, 13, 33, 80, 185, 411, 885, 1862, 3853, 7881, 15993, 32284, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{4}-9 x^{3}+10 x^{2}-5 x +1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 5$$
$$\displaystyle a \! \left(4\right) = 13$$
$$\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n^{2}-2 n +6\right)}{6}, \quad n \geq 5$$
Explicit Closed Form
$$\displaystyle -1-\frac{11 n}{6}-\frac{n^{3}}{6}+2^{n +1}$$

### This specification was found using the strategy pack "Point Placements" and has 32 rules.

Found on January 18, 2022.

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