Av(12354, 12453, 12543)
Counting Sequence
1, 1, 2, 6, 24, 117, 652, 3988, 26112, 180126, 1295090, 9631656, 73676572, 577180996, 4615090192, ...
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 45 rules.
Finding the specification took 245 seconds.
Copy 45 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_{15}\! \left(x , y_{0}, y_{1}\right)+F_{18}\! \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_{12}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{12}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{1}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{17}\! \left(x , y_{0}, 1\right) y_{0}-F_{17}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, y_{0} 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_{21}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}+F_{21}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y_{0}, y_{1}\right)+F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{44}\! \left(x , y_{1}, y_{0}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{21}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{21}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
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) &= F_{26}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{37}\! \left(x , y_{1}, y_{0}, y_{2}\right)+F_{42}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{30}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{3}\! \left(x \right) F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}+F_{27}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{34}\! \left(x , y_{0}, 1, y_{1}, y_{2}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= -\frac{-F_{35}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}, y_{3}\right) y_{0} y_{1}+F_{35}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{2}, y_{3}\right) y_{2}}{y_{0} y_{1}-y_{2}}\\
F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{36}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}, y_{3}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{27}\! \left(x , y_{0}, y_{1} y_{2}, y_{2} y_{3}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x , y_{0}, y_{1}, y_{0} y_{2}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{39}\! \left(x , y_{1}, y_{0}, y_{2}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{40}\! \left(x , y_{0}, 1, y_{1}, y_{2}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= -\frac{-F_{41}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}\right) y_{1}+F_{41}\! \left(x , y_{0}, y_{2}, y_{3}\right)}{y_{1}-1}\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{21}\! \left(x , y_{0}, y_{1}\right)-y_{2} F_{21}\! \left(x , y_{0}, y_{2}\right)}{-y_{2}+y_{1}}\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{43}\! \left(x , y_{0} y_{1}, y_{2}, y_{0}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{41}\! \left(x , y_{1}, y_{2}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
\end{align*}\)
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 43 rules.
Finding the specification took 25961 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_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{5}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= -\frac{-F_{4}\! \left(x , y\right) y +F_{4}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y , 1\right)\\
F_{8}\! \left(x , y , z\right) &= F_{9}\! \left(x , y , y z \right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{19}\! \left(x , y\right)\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right)+F_{5}\! \left(x , y\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x , y , z\right) &= -\frac{-y F_{14}\! \left(x , y , z\right)+F_{14}\! \left(x , 1, z\right)}{-1+y}\\
F_{14}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x , y , z\right) &= \frac{z \left(F_{17}\! \left(x , y , 1\right)-F_{17}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{19}\! \left(x , y\right)\\
F_{18}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , y z \right)\\
F_{19}\! \left(x , y\right) &= y x\\
F_{20}\! \left(x , y , z\right) &= F_{19}\! \left(x , z\right) F_{21}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , z\right)+F_{31}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{22}\! \left(x , 1, y\right)\\
F_{28}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= y x\\
F_{31}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)\\
F_{32}\! \left(x , y , z\right) &= F_{19}\! \left(x , y\right) F_{33}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y , z\right) &= F_{19}\! \left(x , z\right) F_{33}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y , z\right) &= F_{11}\! \left(x , y , z\right)\\
F_{35}\! \left(x , y , z\right) &= z F_{36}\! \left(x , y , z\right)\\
F_{36}\! \left(x , y , z\right) &= F_{37}\! \left(x , z , y\right)\\
F_{38}\! \left(x , y , z\right) &= F_{37}\! \left(x , y , z\right)+F_{39}\! \left(x , y , z\right)\\
F_{38}\! \left(x , y , z\right) &= F_{39}\! \left(x , y , z\right)+F_{40}\! \left(x , y , z\right)\\
F_{39}\! \left(x , y , z\right) &= F_{22}\! \left(x , z , y\right)\\
F_{40}\! \left(x , y , z\right) &= F_{41}\! \left(x , y , z\right)\\
F_{41}\! \left(x , y , z\right) &= F_{15}\! \left(x \right) F_{42}\! \left(x , y , z\right)\\
F_{42}\! \left(x , y , z\right) &= F_{33}\! \left(x , z , y\right)\\
\end{align*}\)