###### Av(2143, 2413)
Counting Sequence
1, 1, 2, 6, 22, 90, 395, 1823, 8741, 43193, 218704, 1129944, 5937728, 31656472, 170892498, ...
Implicit Equation for the Generating Function
$$\displaystyle x^{4} F \left(x \right)^{3}+x^{2} \left(5 x -11\right) F \left(x \right)^{2}+\left(3 x^{2}+10 x -1\right) F \! \left(x \right)-9 x +1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(n +3\right) = \frac{16 \left(2 n +3\right) \left(n +1\right) a \! \left(n \right)}{5 \left(n +5\right) \left(n +4\right)}-\frac{\left(69 n^{2}+313 n +328\right) a \! \left(n +1\right)}{5 \left(n +5\right) \left(n +4\right)}+\frac{2 \left(21 n +55\right) a \! \left(n +2\right)}{5 \left(n +5\right)}, \quad n \geq 3$$
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 "Point And Col Placements Tracked Fusion Req Corrob" and has 23 rules.

Found on April 22, 2021.

Finding the specification took 88 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_{0}\! \left(x \right) F_{16}\! \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_{4}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= -\frac{y \left(F_{9}\! \left(x , 1\right)-F_{9}\! \left(x , y\right)\right)}{-1+y}\\ F_{22}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Point Placements Tracked Fusion Isolated Req Corrob" and has 26 rules.

Found on April 22, 2021.

Finding the specification took 2514 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \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_{10}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= -\frac{y \left(F_{10}\! \left(x , 1\right)-F_{10}\! \left(x , y\right)\right)}{-1+y}\\ F_{25}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Req Corrob" and has 25 rules.

Found on April 27, 2021.

Finding the specification took 4038 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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{18}\! \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_{4}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= -\frac{y \left(F_{10}\! \left(x , 1\right)-F_{10}\! \left(x , y\right)\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 23 rules.

Found on April 27, 2021.

Finding the specification took 2800 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_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= -\frac{y \left(F_{9}\! \left(x , 1\right)-F_{9}\! \left(x , y\right)\right)}{-1+y}\\ F_{22}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ \end{align*}