Av(13425, 13452, 14325, 14352, 14532, 41325, 41352, 41532, 45132)
Counting Sequence
1, 1, 2, 6, 24, 111, 546, 2758, 14110, 72687, 375998, 1950212, 10134024, 52730484, 274647566, ...
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 46 rules.
Finding the specification took 164 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_{3}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \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_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+4 x^{2} F_{7}\! \left(x , y\right) y^{2}+4 y^{2} x^{2}+4 x F_{7}\! \left(x , y\right)^{2} y -5 x F_{7}\! \left(x , y\right) y -y x -F_{7}\! \left(x , y\right)^{2}+2 F_{7}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{30}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{26}\! \left(x , y\right) F_{31}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{30}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{30}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= x^{2} F_{21}\! \left(x , y\right)^{2} y^{2}+2 x^{2} F_{21}\! \left(x , y\right) y^{2}+y^{2} x^{2}+4 x F_{21}\! \left(x , y\right)^{2} y -13 x F_{21}\! \left(x , y\right) y +8 y x -F_{21}\! \left(x , y\right)^{2}+4 F_{21}\! \left(x , y\right)-2\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)^{2} F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= y x\\
F_{27}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= -\frac{-F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\
F_{30}\! \left(x \right) &= x\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31} \left(x \right)^{2} F_{30}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{26}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , 1, y\right)\\
F_{42}\! \left(x , y , z\right) &= -\frac{-F_{43}\! \left(x , y z \right) y +F_{43}\! \left(x , z\right)}{-1+y}\\
F_{43}\! \left(x , y\right) &= -\frac{-F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\
F_{44}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 37 rules.
Finding the specification took 162 seconds.
Copy 37 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_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{26}\! \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) &= -\frac{-F_{8}\! \left(x , y\right) y +F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x^{2} F_{9} \left(x \right)^{2}+2 x^{2} F_{9}\! \left(x \right)+4 x F_{9} \left(x \right)^{2}+x^{2}-13 x F_{9}\! \left(x \right)-F_{9} \left(x \right)^{2}+8 x +4 F_{9}\! \left(x \right)-2\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{24}\! \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_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)^{2} F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , 1, y\right)\\
F_{19}\! \left(x , y , z\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{20}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{17}\! \left(x , y\right) F_{20}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(y x , z\right)\\
F_{8}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{26}\! \left(x \right) &= x\\
F_{27}\! \left(x \right) &= F_{26}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x , 1\right)\\
F_{29}\! \left(x , y\right) &= -\frac{-F_{30}\! \left(x , y\right) y +F_{30}\! \left(x , 1\right)}{-1+y}\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= -\frac{-F_{14}\! \left(x , y\right) y +F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y , 1\right)\\
F_{35}\! \left(x , y , z\right) &= \frac{y z F_{22}\! \left(x , y , z\right)-F_{22}\! \left(x , y , \frac{1}{y}\right)}{y z -1}\\
F_{36}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{29}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 35 rules.
Finding the specification took 708 seconds.
Copy 35 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_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \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_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+4 x^{2} F_{7}\! \left(x , y\right) y^{2}+4 y^{2} x^{2}+4 x F_{7}\! \left(x , y\right)^{2} y -5 x F_{7}\! \left(x , y\right) y -y x -F_{7}\! \left(x , y\right)^{2}+2 F_{7}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{31}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{32}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{31}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= -\frac{-F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)^{2} F_{21}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= y x\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y , 1\right)\\
F_{23}\! \left(x , y , z\right) &= \frac{F_{24}\! \left(x , y , z\right)-F_{24}\! \left(x , y , \frac{1}{y}\right)}{y z -1}\\
F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{20}\! \left(x , z\right) F_{21}\! \left(x , z\right) F_{26}\! \left(x , y , z\right)\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , z\right)+F_{28}\! \left(x , y , z\right)\\
F_{27}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)+F_{7}\! \left(x , y\right)\\
F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y , z\right) F_{7}\! \left(x , y\right)\\
F_{29}\! \left(x , y , z\right) &= F_{30}\! \left(y x , z\right)\\
F_{30}\! \left(x , y\right) &= -\frac{-y F_{20}\! \left(x , y\right)+F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{31}\! \left(x \right) &= x\\
F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{32} \left(x \right)^{2} F_{31}\! \left(x \right)\\
\end{align*}\)