Av(12345, 12354, 12435)
Counting Sequence
1, 1, 2, 6, 24, 117, 653, 4010, 26427, 183888, 1335999, 10053592, 77894889, 618573774, 5016772752, ...
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Expand Verified" and has 44 rules.
Finding the specification took 158 seconds.
Copy 44 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_{4}\! \left(x \right)+F_{42}\! \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_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , 1, y\right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y z , z\right)\\
F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{13}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{33}\! \left(x , y , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{14}\! \left(x , y , z\right)\\
F_{14}\! \left(x , y , z\right) &= -\frac{F_{9}\! \left(x , 1, z\right) z -F_{9}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{21}\! \left(x \right)\\
F_{16}\! \left(x , y , z\right) &= \frac{F_{17}\! \left(x , y\right) y -F_{17}\! \left(x , z\right) z}{-z +y}\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= -\frac{-F_{17}\! \left(x , y\right) y +F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{23}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y , 1\right)\\
F_{25}\! \left(x , y , z\right) &= -\frac{-F_{26}\! \left(x , y , z\right) z +F_{26}\! \left(x , y , 1\right)}{-1+z}\\
F_{26}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y , z\right)+F_{28}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)+F_{33}\! \left(x , y , z\right)\\
F_{27}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{26}\! \left(x , y , z\right)\\
F_{28}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{29}\! \left(x , y , z\right)\\
F_{29}\! \left(x , y , z\right) &= -\frac{F_{30}\! \left(x , 1, z\right) z -F_{30}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{30}\! \left(x , y , z\right) &= F_{26}\! \left(x , y z , z\right)\\
F_{31}\! \left(x , y , z\right) &= F_{21}\! \left(x \right) F_{32}\! \left(x , y , z\right)\\
F_{32}\! \left(x , y , z\right) &= -\frac{-F_{10}\! \left(x , y , z\right) z +F_{10}\! \left(x , y , 1\right)}{-1+z}\\
F_{33}\! \left(x , y , z\right) &= F_{21}\! \left(x \right) F_{25}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= -\frac{-F_{37}\! \left(x , y\right) y +F_{37}\! \left(x , 1\right)}{-1+y}\\
F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{30}\! \left(x , 1, y\right)\\
F_{40}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{42}\! \left(x \right) &= F_{21}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 33 rules.
Finding the specification took 1043 seconds.
Copy 33 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_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \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_{18}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , 1, y\right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y z , z\right)\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{13}\! \left(x , z\right)\\
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_{15}\! \left(x , y , z\right)+F_{19}\! \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) &= -\frac{F_{9}\! \left(x , 1, z\right) z -F_{9}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{15}\! \left(x , y , z\right) &= \frac{F_{16}\! \left(x , y\right) y -F_{16}\! \left(x , z\right) z}{-z +y}\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y , 1\right)\\
F_{17}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x , y , z\right) &= -\frac{-F_{20}\! \left(x , y , z\right) z +F_{20}\! \left(x , y , 1\right)}{-1+z}\\
F_{20}\! \left(x , y , z\right) &= F_{18}\! \left(x \right) F_{21}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{22}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{13}\! \left(x , y\right) F_{21}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y , z\right) &= -\frac{F_{24}\! \left(x , 1, z\right) z -F_{24}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y z , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{21}\! \left(x , y , z\right)\\
F_{26}\! \left(x , y , z\right) &= -\frac{-z F_{17}\! \left(x , y , z\right)+F_{17}\! \left(x , y , 1\right)}{-1+z}\\
F_{27}\! \left(x , y\right) &= -\frac{-F_{28}\! \left(x , y\right) y +F_{28}\! \left(x , 1\right)}{-1+y}\\
F_{28}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{30}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{24}\! \left(x , 1, y\right)\\
F_{31}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\
\end{align*}\)