Av(12543, 13542, 21543, 23541, 31542, 32541, 41532, 42531)
Counting Sequence
1, 1, 2, 6, 24, 112, 568, 3030, 16726, 94656, 545968, 3196948, 18951262, 113495516, 685607382, ...
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 39 rules.
Finding the specification took 272 seconds.
Copy 39 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , 1, y\right)\\
F_{21}\! \left(x , y , z\right) &= -\frac{-F_{22}\! \left(x , y , z\right) y +F_{22}\! \left(x , 1, z\right)}{-1+y}\\
F_{22}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , z\right)+F_{23}\! \left(x , y , z\right)+F_{25}\! \left(x , y , z\right)+F_{27}\! \left(x , z\right)+F_{28}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y , z\right) &= F_{12}\! \left(x \right) F_{24}\! \left(x , y , z\right)\\
F_{24}\! \left(x , y , z\right) &= -\frac{-F_{22}\! \left(x , y , z\right) y +F_{22}\! \left(x , 1, z\right)}{-1+y}\\
F_{25}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right) F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= y x\\
F_{27}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{28}\! \left(x , y , z\right) &= F_{12}\! \left(x \right) F_{29}\! \left(x , y , z\right)\\
F_{29}\! \left(x , y , z\right) &= -\frac{-F_{22}\! \left(x , y , z\right) z +F_{22}\! \left(x , y , 1\right)}{-1+z}\\
F_{30}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= -\frac{-F_{18}\! \left(x , y\right) y +F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{32}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{36}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 38 rules.
Finding the specification took 50 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{33}\! \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 \right)\\
F_{10}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , 1, y\right)\\
F_{20}\! \left(x , y , z\right) &= -\frac{-F_{21}\! \left(x , y , z\right) y +F_{21}\! \left(x , 1, z\right)}{-1+y}\\
F_{21}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , z\right)+F_{22}\! \left(x , y , z\right)+F_{24}\! \left(x , y , z\right)+F_{26}\! \left(x , z\right)+F_{27}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{11}\! \left(x \right) F_{23}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y , z\right) &= -\frac{-F_{21}\! \left(x , y , z\right) y +F_{21}\! \left(x , 1, z\right)}{-1+y}\\
F_{24}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= y x\\
F_{26}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\
F_{27}\! \left(x , y , z\right) &= F_{11}\! \left(x \right) F_{28}\! \left(x , y , z\right)\\
F_{28}\! \left(x , y , z\right) &= -\frac{-F_{21}\! \left(x , y , z\right) z +F_{21}\! \left(x , y , 1\right)}{-1+z}\\
F_{29}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= -\frac{-F_{17}\! \left(x , y\right) y +F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{31}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{33}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{35}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)