###### Av(1342, 1423)
Generating Function
$$\displaystyle -\frac{x}{2}+\frac{3}{2}-\frac{\sqrt{x^{2}-6 x +1}}{2}$$
Counting Sequence
1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, ...
Implicit Equation for the Generating Function
$$\displaystyle F \left(x \right)^{2}+\left(x -3\right) F \! \left(x \right)+2 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2$$
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 Row Placements Tracked Fusion" and has 14 rules.

Found on April 26, 2021.

Finding the specification took 100 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 14 equations to clipboard:
\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_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{6}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{6}\! \left(x \right) &= 0\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x , y\right) &= -\frac{y \left(F_{5}\! \left(x , 1\right)-F_{5}\! \left(x , y\right)\right)}{y -1}\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{4}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{5}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 14 rules.

Found on April 26, 2021.

Finding the specification took 139 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 14 equations to clipboard:
\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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{7}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= y x\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{6}\! \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 16 rules.

Found on April 26, 2021.

Finding the specification took 130 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 16 equations to clipboard:
\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_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= 0\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{8}\! \left(x , 1\right)-F_{8}\! \left(x , y\right)\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ \end{align*}