Av(12354, 13254, 21354)
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 Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 46 rules.
Finding the specification took 106 seconds.
Copy 46 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_{4}\! \left(x \right)+F_{44}\! \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) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{42}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y\right) &= -\frac{-F_{6}\! \left(x , y\right) y +F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-F_{12}\! \left(x , y\right) y +F_{12}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{40}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , 1, y\right)\\
F_{17}\! \left(x , y , z\right) &= -\frac{-F_{18}\! \left(x , y , z\right) y +F_{18}\! \left(x , 1, z\right)}{-1+y}\\
F_{18}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)+F_{32}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{20}\! \left(x , y , z\right) &= -\frac{-F_{18}\! \left(x , y , z\right) y +F_{18}\! \left(x , 1, z\right)}{-1+y}\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{22}\! \left(x , y , z\right) &= -\frac{-F_{23}\! \left(x , y , z\right) y +F_{23}\! \left(x , 1, z\right)}{-1+y}\\
F_{24}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , y z \right)\\
F_{24}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)+F_{28}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , y z \right)\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x , y , z\right) &= -\frac{-F_{14}\! \left(x , y z \right) z +F_{14}\! \left(x , y\right)}{-1+z}\\
F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y , z\right) F_{30}\! \left(x , y\right)\\
F_{29}\! \left(x , y , z\right) &= -\frac{-F_{24}\! \left(x , y , z\right) z +F_{24}\! \left(x , y , 1\right)}{-1+z}\\
F_{30}\! \left(x , y\right) &= y x\\
F_{31}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\
F_{32}\! \left(x , y , z\right) &= F_{30}\! \left(x , y\right) F_{33}\! \left(x , y , z\right)\\
F_{33}\! \left(x , y , z\right) &= \frac{y F_{34}\! \left(x , y , 1\right)-F_{34}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{34}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , y z \right)\\
F_{35}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{37}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{38}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{24}\! \left(x , y , 1\right)\\
F_{42}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{34}\! \left(x , y , 1\right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \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 40 rules.
Finding the specification took 1253 seconds.
Copy 40 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_{39}\! \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_{4}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= -\frac{-F_{7}\! \left(x , y\right) y +F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{10}\! \left(x , y\right) &= -\frac{-F_{11}\! \left(x , y\right) y +F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{37}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{33}\! \left(x , y\right)+F_{34}\! \left(x , y\right)+F_{36}\! \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_{17}\! \left(x , y , z\right) y +F_{17}\! \left(x , 1, z\right)}{-1+y}\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{29}\! \left(x , y , z\right)+F_{32}\! \left(x , z , y\right)\\
F_{19}\! \left(x , y , z\right) &= -\frac{-F_{17}\! \left(x , y , z\right) y +F_{17}\! \left(x , 1, z\right)}{-1+y}\\
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_{22}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)+F_{24}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)\\
F_{23}\! \left(x , y , z\right) &= \frac{F_{13}\! \left(x , y\right) y -F_{13}\! \left(x , z\right) z}{-z +y}\\
F_{24}\! \left(x , y , z\right) &= \frac{F_{25}\! \left(x , y , 1\right) y -F_{25}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y , y z \right)\\
F_{26}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right) F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= y x\\
F_{28}\! \left(x , y , z\right) &= F_{22}\! \left(x , z , y\right) F_{27}\! \left(x , y\right)\\
F_{29}\! \left(x , y , z\right) &= \frac{y F_{30}\! \left(x , y , 1\right)-F_{30}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{30}\! \left(x , y , z\right) &= F_{31}\! \left(x , y , y z \right)\\
F_{31}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{27}\! \left(x , y\right)\\
F_{32}\! \left(x , y , z\right) &= F_{18}\! \left(x , z , y\right) F_{27}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= -\frac{-F_{13}\! \left(x , y\right) y +F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= -\frac{-y F_{35}\! \left(x , y\right)+F_{35}\! \left(x , 1\right)}{-1+y}\\
F_{35}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{25}\! \left(x , y , 1\right)\\
F_{38}\! \left(x , y\right) &= F_{30}\! \left(x , y , 1\right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
\end{align*}\)