Av(12345, 12435, 12453, 13245)
Counting Sequence
1, 1, 2, 6, 24, 116, 635, 3791, 24134, 161486, 1124514, 8091104, 59831778, 452835816, 3496384939, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 43 rules.
Finding the specification took 104 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_{37}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}, 1\right)\\
F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{36}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, y_{1}\right)\\
F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{1}\right) F_{12}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}\right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{1}\right) F_{12}\! \left(x , y_{0}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{14}\! \left(x , y_{0}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{16}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{16}\! \left(x , y_{1}, y_{2}\right)}{y_{0}-1}\\
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_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{16}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{19}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{20}\! \left(x , 1, y_{1}\right) y_{1}-F_{20}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0} y_{1}, y_{1}\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_{24}\! \left(x , y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}\right) &= F_{25}\! \left(x , y_{0}, 1\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{9}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{28}\! \left(x , 1, y_{1}\right) y_{1}-F_{28}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right) F_{37}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{31}\! \left(x , 1, y_{1}\right) y_{1}-F_{31}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{33}\! \left(x , y_{0}, y_{1}\right) y_{1}+F_{33}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{1}\right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= x\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x \right) F_{39}\! \left(x , y_{0}, y_{1}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{40}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{32}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{32}\! \left(x , y_{1}, y_{2}\right)}{y_{0}-1}\\
F_{41}\! \left(x , y_{0}\right) &= F_{37}\! \left(x \right) F_{42}\! \left(x , y_{0}\right)\\
F_{42}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{4}\! \left(x , y_{0}\right)+F_{4}\! \left(x , 1\right)}{y_{0}-1}\\
\end{align*}\)
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 37 rules.
Finding the specification took 6905 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_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{5}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{35}\! \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_{11}\! \left(x , y , z\right) F_{20}\! \left(x , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{33}\! \left(x , y , z\right)+F_{8}\! \left(x , z\right)\\
F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right)\\
F_{16}\! \left(x , y , z\right) &= -\frac{F_{17}\! \left(x , 1, z\right) z -F_{17}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{20}\! \left(x , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)+F_{22}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{20}\! \left(x , z\right)\\
F_{20}\! \left(x , y\right) &= y x\\
F_{21}\! \left(x , y , z\right) &= -\frac{-F_{9}\! \left(x , y , z\right) y +F_{9}\! \left(x , 1, z\right)}{-1+y}\\
F_{22}\! \left(x , y , z\right) &= F_{16}\! \left(x , y z , z\right)\\
F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z , z\right)\\
F_{24}\! \left(x , y , z\right) &= -\frac{F_{25}\! \left(x , 1, z\right) z -F_{25}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\
F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y z , z\right)\\
F_{26}\! \left(x , y , z\right) &= -\frac{-F_{27}\! \left(x , y , z\right) z +F_{27}\! \left(x , y , 1\right)}{-1+z}\\
F_{27}\! \left(x , y , z\right) &= F_{28}\! \left(x , y , z\right) F_{32}\! \left(x \right)\\
F_{29}\! \left(x , y , z\right) &= F_{28}\! \left(x , y z , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{25}\! \left(x , y , z\right)+F_{30}\! \left(x , y , z\right)\\
F_{30}\! \left(x , y , z\right) &= F_{31}\! \left(x , y z , z\right)\\
F_{31}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{28}\! \left(x , y , z\right)\\
F_{32}\! \left(x \right) &= x\\
F_{33}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y\right) &= F_{17}\! \left(x , 1, y\right)\\
F_{35}\! \left(x , y\right) &= F_{25}\! \left(x , 1, y\right)\\
F_{36}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)