###### Av(1342, 3142)

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

\(\displaystyle a \! \left(0\right) = 1\)

\(\displaystyle a \! \left(1\right) = 1\)

\(\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)

\(\displaystyle a \! \left(1\right) = 1\)

\(\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)

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 "Row And Col Placements" and has 7 rules.

Found on April 26, 2021.Finding the specification took 1 seconds.

\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4} \left(x \right)^{2} F_{3}\! \left(x \right)\\
\end{align*}\)

### This specification was found using the strategy pack "Insertion Row And Col Placements" and has 15 rules.

Found on April 26, 2021.Finding the specification took 3 seconds.

\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)

### This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob" and has 8 rules.

Found on April 26, 2021.Finding the specification took 2 seconds.

\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{4}\! \left(x \right)\\
\end{align*}\)