Av(12345, 12354, 12453)
Counting Sequence
1, 1, 2, 6, 24, 117, 652, 3986, 26050, 178963, 1277820, 9407127, 70990882, 546790230, 4284188730, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 51 rules.
Finding the specification took 51 seconds.
Copy 51 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_{17}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{17}\! \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_{35}\! \left(x , y\right)+F_{37}\! \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_{18}\! \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_{17}\! \left(x \right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{-F_{10}\! \left(x , y , z\right) z +F_{10}\! \left(x , y , 1\right)}{-1+z}\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x \right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= \frac{F_{20}\! \left(x , y\right) y -F_{20}\! \left(x , z\right) z}{-z +y}\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= -\frac{-F_{20}\! \left(x , y\right) y +F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{24}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{16}\! \left(x , y , 1\right)\\
F_{26}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y , 1\right)\\
F_{28}\! \left(x , y , z\right) &= -\frac{-F_{29}\! \left(x , y , z\right) z +F_{29}\! \left(x , y , 1\right)}{-1+z}\\
F_{29}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y , z\right)+F_{30}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)+F_{34}\! \left(x , y , z\right)\\
F_{30}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{29}\! \left(x , y , z\right)\\
F_{31}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{32}\! \left(x , y , z\right)\\
F_{32}\! \left(x , y , z\right) &= -\frac{F_{33}\! \left(x , 1, z\right) z -F_{33}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{33}\! \left(x , y , z\right) &= F_{29}\! \left(x , y z , z\right)\\
F_{34}\! \left(x , y , z\right) &= F_{17}\! \left(x \right) F_{28}\! \left(x , y , z\right)\\
F_{35}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= -\frac{-F_{6}\! \left(x , y\right) y +F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{37}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{38}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{17}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\
F_{42}\! \left(x \right) &= F_{17}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{36}\! \left(x , 1\right)\\
F_{44}\! \left(x \right) &= F_{17}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x , 1\right)\\
F_{46}\! \left(x , y\right) &= -\frac{-y F_{47}\! \left(x , y\right)+F_{47}\! \left(x , 1\right)}{-1+y}\\
F_{47}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{33}\! \left(x , 1, y\right)\\
F_{50}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{46}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 44 rules.
Finding the specification took 1784 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_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \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_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y\right) &= -\frac{-F_{13}\! \left(x , y\right) y +F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y , 1\right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{-F_{17}\! \left(x , y , z\right) z +F_{17}\! \left(x , y , 1\right)}{-1+z}\\
F_{17}\! \left(x , y , z\right) &= F_{14}\! \left(x \right) F_{18}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y , z\right)+F_{19}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{18}\! \left(x , y , z\right)\\
F_{20}\! \left(x , y , z\right) &= -\frac{F_{21}\! \left(x , 1, z\right) z -F_{21}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y z , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{11}\! \left(x , z\right) F_{18}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y , z\right) &= -\frac{-F_{24}\! \left(x , y , z\right) z +F_{24}\! \left(x , y , 1\right)}{-1+z}\\
F_{24}\! \left(x , y , z\right) &= F_{14}\! \left(x \right) F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)+F_{27}\! \left(x , y , z\right)+F_{30}\! \left(x , y , z\right)\\
F_{26}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{25}\! \left(x , y , z\right)\\
F_{27}\! \left(x , y , z\right) &= -\frac{F_{28}\! \left(x , 1, z\right) z -F_{28}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y z , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{11}\! \left(x , z\right) F_{25}\! \left(x , y , z\right)\\
F_{30}\! \left(x , y , z\right) &= \frac{F_{31}\! \left(x , y\right) y -F_{31}\! \left(x , z\right) z}{-z +y}\\
F_{31}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{23}\! \left(x , y , 1\right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\
F_{34}\! \left(x , y\right) &= -\frac{-F_{35}\! \left(x , y\right) y +F_{35}\! \left(x , 1\right)}{-1+y}\\
F_{35}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{37}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{21}\! \left(x , 1, y\right)\\
F_{38}\! \left(x , y\right) &= -\frac{-y F_{39}\! \left(x , y\right)+F_{39}\! \left(x , 1\right)}{-1+y}\\
F_{39}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{28}\! \left(x , 1, y\right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x , 1\right)\\
F_{43}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\
\end{align*}\)