Av(12453, 13452, 14352, 21453, 23451, 24351, 31452, 32451)
Counting Sequence
1, 1, 2, 6, 24, 112, 570, 3062, 17066, 97700, 570862, 3389812, 20394106, 124034760, 761292536, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 43 rules.
Finding the specification took 105 seconds.
Copy 43 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_{14}\! \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_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= -\frac{-F_{17}\! \left(x , y\right) y +F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{18}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{20}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= y x\\
F_{22}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= -\frac{-F_{10}\! \left(x , y\right) y +F_{10}\! \left(x , 1\right)}{-1+y}\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\
F_{28}\! \left(x , y\right) &= -\frac{-F_{17}\! \left(x , y\right) y +F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{29}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y , 1\right)\\
F_{31}\! \left(x , y , z\right) &= -\frac{-F_{32}\! \left(x , y , z\right) z +F_{32}\! \left(x , y , 1\right)}{-1+z}\\
F_{32}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , z\right)+F_{20}\! \left(x , y\right)+F_{33}\! \left(x , y , z\right)+F_{36}\! \left(x , y , z\right)+F_{37}\! \left(x , y , z\right)\\
F_{33}\! \left(x , y , z\right) &= F_{14}\! \left(x \right) F_{34}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y , z\right) &= -\frac{-F_{35}\! \left(x , y , z\right) y +F_{35}\! \left(x , 1, z\right)}{-1+y}\\
F_{35}\! \left(x , y , z\right) &= \frac{y F_{11}\! \left(x , y\right)-z F_{11}\! \left(x , z\right)}{-z +y}\\
F_{36}\! \left(x , y , z\right) &= F_{21}\! \left(x , z\right) F_{32}\! \left(x , y , z\right)\\
F_{37}\! \left(x , y , z\right) &= F_{14}\! \left(x \right) F_{31}\! \left(x , y , z\right)\\
F_{38}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 80 rules.
Finding the specification took 2460 seconds.
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Copy 80 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_{5}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{67}\! \left(x , y\right)+F_{68}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{5}\! \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_{24}\! \left(x , y\right)+F_{65}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{5}\! \left(x \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_{18}\! \left(x , y\right)+F_{63}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= -\frac{-F_{20}\! \left(x , y\right) y +F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)+F_{29}\! \left(x , y\right)+F_{61}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{5}\! \left(x \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_{13}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{25}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\
F_{27}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= y x\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= -\frac{-F_{31}\! \left(x , y\right) y +F_{31}\! \left(x , 1\right)}{-1+y}\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{6}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{42}\! \left(x , y\right)+F_{6}\! \left(x \right)+F_{60}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{5}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{46}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{53}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x , y\right)+F_{49}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= y x\\
F_{53}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x , 1\right)\\
F_{56}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{58}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= -\frac{-y F_{56}\! \left(x , y\right)+F_{56}\! \left(x , 1\right)}{-1+y}\\
F_{59}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{56}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{62}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{64}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{68}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{69}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x , 1\right)\\
F_{71}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{5}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{31}\! \left(x , 1\right)\\
F_{74}\! \left(x \right) &= F_{5}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x , 1\right)\\
F_{77}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x , 1\right)\\
F_{79}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{5}\! \left(x \right)\\
\end{align*}\)