###### Av(1342, 2314)
Generating Function
$$\displaystyle \frac{-3 x^{2}-x^{2} \sqrt{-4 x +1}+5 x -1}{4 x^{3}-8 x^{2}+6 x -1}$$
Counting Sequence
1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(4 x^{3}-8 x^{2}+6 x -1\right) F \left(x \right)^{2}+\left(6 x^{2}-10 x +2\right) F \! \left(x \right)+x^{2}+4 x -1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(n +4\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{2+n}-\frac{4 \left(8 n +7\right) a \! \left(2+n \right)}{2+n}+\frac{12 \left(3 n +2\right) a \! \left(n +1\right)}{2+n}+\frac{2 \left(5 n +7\right) a \! \left(n +3\right)}{2+n}, \quad n \geq 4$$
Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point $$(i, j)$$ represents how many permutations have value $$j$$ at index $$i$$ (darker = more).

### This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 32 rules.

Found on April 26, 2021.

Finding the specification took 800 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{17}\! \left(x , y\right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 38 rules.

Found on April 26, 2021.

Finding the specification took 1603 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_{16}\! \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_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= -\frac{-y F_{12}\! \left(x , y\right)+F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\ F_{28}\! \left(x , y\right) &= -\frac{-y F_{29}\! \left(x , y\right)+F_{29}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{21}\! \left(x , y\right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{31} \left(x \right)^{2} F_{16}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 38 rules.

Found on April 26, 2021.

Finding the specification took 1571 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_{18}\! \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_{18}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{35}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{19}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= 0\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{9}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{14}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= -\frac{-y F_{29}\! \left(x , y\right)+F_{29}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{29}\! \left(x , y\right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32} \left(x \right)^{2} F_{18}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x , 1\right)\\ F_{37}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{29}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Req Corrob" and has 32 rules.

Found on April 26, 2021.

Finding the specification took 1044 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{23}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x , 1\right)\\ F_{31}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{23}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion" and has 33 rules.

Found on April 26, 2021.

Finding the specification took 922 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_{13}\! \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_{13}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{18}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\ \end{align*}