Av(1342, 2341)
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Generating Function
\(\displaystyle -\frac{x}{2}+\frac{3}{2}-\frac{\sqrt{x^{2}-6 x +1}}{2}\)
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Counting Sequence
1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, ...
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Implicit Equation for the Generating Function
\(\displaystyle F \left(x \right)^{2}+\left(x -3\right) F \! \left(x \right)+2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)
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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 "Col Placements Tracked Fusion Isolated" and has 14 rules.

Found on April 21, 2021.

Finding the specification took 2 seconds.

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

This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion Req Corrob" and has 22 rules.

Found on April 26, 2021.

Finding the specification took 19 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{19}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Col Placements Tracked Fusion Isolated" and has 23 rules.

Found on April 21, 2021.

Finding the specification took 4 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= -\frac{y \left(F_{16}\! \left(x , 1\right)-F_{16}\! \left(x , y\right)\right)}{-1+y}\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{19}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x , y\right) &= -\frac{y \left(F_{11}\! \left(x , 1\right)-F_{11}\! \left(x , y\right)\right)}{-1+y}\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion" and has 15 rules.

Found on April 26, 2021.

Finding the specification took 9 seconds.

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