Av(13254, 13524, 13542)
The requested set is not in the database, but a symmetry of it is.
Counting Sequence
1, 1, 2, 6, 24, 117, 652, 3988, 26112, 180126, 1295090, 9631656, 73676572, 577180996, 4615090192, ...
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 Expand Verified" and has 30 rules.
Found on November 06, 2021.Finding the specification took 816 seconds.
Copy 30 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_{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_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right)+F_{5}\! \left(x , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right)\\
F_{16}\! \left(x , y , z\right) &= \frac{y z \left(F_{12}\! \left(x , y , z\right)-F_{12}\! \left(x , \frac{1}{z}, z\right)\right)}{y z -1}\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{9}\! \left(x , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{22}\! \left(x , z , y\right)\\
F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x , y , z\right) &= \frac{y z F_{19}\! \left(x , y , z\right)-F_{19}\! \left(x , \frac{1}{z}, z\right)}{y z -1}\\
F_{22}\! \left(x , y , z\right) &= F_{19}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\
F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z , z\right)\\
F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= \frac{y F_{26}\! \left(x , y , 1\right)-z F_{26}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , y z \right)\\
F_{12}\! \left(x , y , z\right) &= F_{27}\! \left(x , y z , z\right)\\
F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{12}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 38 rules.
Found on July 27, 2021.Finding the specification took 5394 seconds.
Copy 38 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_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{9}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{11}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{8}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\
F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}, 1\right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{21}\! \left(x , y_{0}, 1, y_{1}\right)-F_{21}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{1}\right)}{-1+y_{0}}\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{1}, y_{0}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}, y_{0}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, 1\right)-y_{1} F_{15}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{15}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 32 rules.
Found on November 06, 2021.Finding the specification took 3416 seconds.
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