Av(12345, 12435, 12453, 14235, 14253, 14523, 41235, 41253, 41523, 45123)
Counting Sequence
1, 1, 2, 6, 24, 110, 530, 2597, 12796, 63156, 311826, 1539461, 7598492, 37496186, 184997956, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 28 rules.
Finding the specification took 1 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_{19}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{19}\! \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) &= -\frac{-F_{7}\! \left(x , y\right) y +F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= -\frac{-F_{8}\! \left(x , y\right) y +F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{12}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , 1, y\right)\\
F_{15}\! \left(x , y , z\right) &= -\frac{-F_{16}\! \left(x , y , z\right) y +F_{16}\! \left(x , 1, z\right)}{-1+y}\\
F_{16}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\
F_{17}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{16}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{15}\! \left(x , y , z\right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\
F_{22}\! \left(x , y\right) &= -\frac{-F_{23}\! \left(x , y\right) y +F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , 1, y\right)\\
F_{26}\! \left(x , y , z\right) &= -\frac{-F_{7}\! \left(x , y z \right) y +F_{7}\! \left(x , z\right)}{-1+y}\\
F_{27}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{22}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 19 rules.
Finding the specification took 8 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= -\frac{-F_{7}\! \left(x , y\right) y +F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= -\frac{-F_{8}\! \left(x , y\right) y +F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= y x F_{8}\! \left(x , y\right)^{2}+1\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , 1, y\right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{-F_{7}\! \left(x , y z \right) y +F_{7}\! \left(x , z\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\
\end{align*}\)