Av(1234, 246135, 256134, 346125, 356124, 456123)
Counting Sequence
1, 1, 2, 6, 23, 103, 508, 2656, 14381, 79533, 445638, 2517738, 14301411, 81529971, 465953088, ...
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 Req Corrob" and has 28 rules.
Found on July 20, 2021.Finding the specification took 92 seconds.
Copy 28 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_{21}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , 1, y\right)\\
F_{7}\! \left(x , y , z\right) &= F_{8}\! \left(x , y z , z\right)\\
F_{8}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{19}\! \left(x , y , z\right)+F_{9}\! \left(x , y , z\right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{13}\! \left(x , y\right)\\
F_{10}\! \left(x , y , z\right) &= \frac{z F_{11}\! \left(x , y , z\right)-F_{11}\! \left(x , y , 1\right)}{-1+z}\\
F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{14}\! \left(x , y , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{15}\! \left(x , y , z\right)\\
F_{15}\! \left(x , y , z\right) &= \frac{-z F_{16}\! \left(x , 1, z\right)+y F_{16}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{16}\! \left(x , y , z\right) &= F_{11}\! \left(x , y z , z\right)\\
F_{17}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{18}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= \frac{-z F_{7}\! \left(x , 1, z\right)+y F_{7}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right) F_{21}\! \left(x \right)\\
F_{20}\! \left(x , y , z\right) &= \frac{z F_{8}\! \left(x , y , z\right)-F_{8}\! \left(x , y , 1\right)}{-1+z}\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\
F_{24}\! \left(x , y\right) &= \frac{y F_{25}\! \left(x , y\right)-F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{24}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion" and has 29 rules.
Found on July 20, 2021.Finding the specification took 86 seconds.
Copy 29 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_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{22}\! \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_{8}\! \left(x , 1, y\right)\\
F_{8}\! \left(x , y , z\right) &= F_{9}\! \left(x , y z , z\right)\\
F_{9}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{14}\! \left(x , y\right)\\
F_{11}\! \left(x , y , z\right) &= -\frac{-z F_{12}\! \left(x , y , z\right)+F_{12}\! \left(x , y , 1\right)}{-1+z}\\
F_{12}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)\\
F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y , z\right) &= F_{14}\! \left(x , z\right) F_{16}\! \left(x , y , z\right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{z F_{17}\! \left(x , 1, z\right)-y F_{17}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{17}\! \left(x , y , z\right) &= F_{12}\! \left(x , y z , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{14}\! \left(x , z\right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= -\frac{z F_{8}\! \left(x , 1, z\right)-y F_{8}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{22}\! \left(x \right)\\
F_{21}\! \left(x , y , z\right) &= -\frac{-z F_{9}\! \left(x , y , z\right)+F_{9}\! \left(x , y , 1\right)}{-1+z}\\
F_{22}\! \left(x \right) &= x\\
F_{23}\! \left(x \right) &= F_{22}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\
F_{25}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\
F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{25}\! \left(x , y\right)\\
\end{align*}\)