Av(12354, 12453, 12543, 13254, 23154)
Counting Sequence
1, 1, 2, 6, 24, 115, 619, 3614, 22425, 145949, 987202, 6893335, 49438329, 362740391, 2714363788, ...
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 42 rules.
Finding the specification took 2417 seconds.
Copy 42 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 , 1\right)\\
F_{4}\! \left(x , y_{0}\right) &= F_{5}\! \left(x \right) F_{6}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= -\frac{-F_{4}\! \left(x , y_{0}\right) y_{0}+F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{8}\! \left(x , y_{0}\right) &= F_{9}\! \left(x , y_{0}, 1\right)\\
F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}\right) F_{12}\! \left(x , y_{0}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{19}\! \left(x , y_{0}, y_{1}\right)+F_{39}\! \left(x , y_{1}, y_{0}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{14}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{14}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{1}\right) F_{5}\! \left(x \right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{1}, y_{0}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{18}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{1}, y_{0}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{1}, y_{0}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{1}, y_{0}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , y_{0}, y_{1}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{1}, y_{0}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{24}\! \left(x , y_{0}, 1\right) y_{0}-F_{24}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{1}, y_{0}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{27}\! \left(x , y_{0}, 1\right) y_{0}-F_{27}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{29}\! \left(x , y_{1}, y_{0}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{30}\! \left(x , y_{0}, 1\right) y_{0}-F_{30}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}\right) F_{31}\! \left(x , y_{0}, y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{1}, y_{0}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{1}\right) F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{1}\right) F_{36}\! \left(x , y_{0}\right)\\
F_{36}\! \left(x , y_{0}\right) &= F_{37}\! \left(x , y_{0}\right)\\
F_{37}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{1}\right) F_{36}\! \left(x , y_{0}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}\right) F_{40}\! \left(x , y_{1}, y_{0}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= F_{41}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{15}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}+F_{15}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 34 rules.
Finding the specification took 22 seconds.
Copy 34 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= -\frac{-F_{5}\! \left(x , y_{0}\right) y_{0}+F_{5}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , 1, y_{0}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{1}, y_{0}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{15}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{15}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , y_{1}, y_{0}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y_{0}, y_{1}\right)+F_{21}\! \left(x , y_{0}, y_{1}\right)+F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{1}, y_{0}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{20}\! \left(x , y_{0}, 1\right) y_{0}-F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{23}\! \left(x , y_{0}, 1\right) y_{0}-F_{23}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{26}\! \left(x , y_{0}, 1\right) y_{0}-F_{26}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{28}\! \left(x , y_{0}, 1\right) y_{0}-F_{28}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{15}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}+F_{15}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 44 rules.
Finding the specification took 42 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= -\frac{-F_{5}\! \left(x , y_{0}\right) y_{0}+F_{5}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , 1, y_{0}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y_{0}, y_{1}\right)+F_{22}\! \left(x , y_{0}, y_{1}\right)+F_{43}\! \left(x , y_{1}, y_{0}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{15}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{15}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{21}\! \left(x , y_{1}, y_{0}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{18}\! \left(x , y_{1}, y_{0}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{15}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}+F_{15}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{41}\! \left(x , y_{1}, y_{0}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{F_{27}\! \left(x , 1, y_{1}, y_{2}\right) y_{1} y_{2}-F_{27}\! \left(x , \frac{y_{0}}{y_{1} y_{2}}, y_{1}, y_{2}\right) y_{0}}{-y_{1} y_{2}+y_{0}}\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{29}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}\right) y_{1} y_{3}-F_{29}\! \left(x , y_{0}, y_{1} y_{2}, \frac{1}{y_{1}}\right)}{y_{1} y_{3}-1}\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} y_{2} F_{33}\! \left(x , y_{0} y_{1}, y_{2}\right)-F_{33}\! \left(x , y_{0} y_{1}, \frac{1}{y_{0}}\right)}{y_{0} y_{2}-1}\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{36}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{38}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0} y_{2}-F_{38}\! \left(x , y_{0} y_{1}, \frac{1}{y_{0}}\right)}{y_{0} y_{2}-1}\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{39}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{40}\! \left(x , y_{0}, 1\right)-y_{1} F_{40}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= F_{29}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= F_{42}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{39}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
\end{align*}\)