Av(12345, 12435, 12453, 13245, 31245)
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 "Row Placements Tracked Fusion" and has 37 rules.
Finding the specification took 90 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_{32}\! \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_{35}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{14}\! \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_{1}\! \left(x \right)+F_{33}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x , y_{0}, y_{1}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}, y_{1}\right)\\
F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}+F_{11}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\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_{30}\! \left(x , y_{0}, y_{1}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, y_{1}\right) F_{14}\! \left(x , y_{0}\right)\\
F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{0}, y_{1}\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_{20}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}\right) F_{18}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{19}\! \left(x , 1, y_{1}\right) y_{1}-F_{19}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{21}\! \left(x , y_{0}, y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{22}\! \left(x , 1, y_{1}\right) y_{1}-F_{22}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{24}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{25}\! \left(x , 1, y_{1}\right) y_{1}-F_{25}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{27}\! \left(x , y_{1}\right)\\
F_{27}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{0}\right)\\
F_{28}\! \left(x , y_{0}\right) &= F_{29}\! \left(x , y_{0}, 1\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , y_{0}, y_{1}\right) F_{32}\! \left(x \right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{12}\! \left(x , y_{0}, y_{1}\right) y_{1}+F_{12}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{32}\! \left(x \right) &= x\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{35}\! \left(x , y_{0}\right) &= F_{32}\! \left(x \right) F_{36}\! \left(x , y_{0}\right)\\
F_{36}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{4}\! \left(x , y_{0}\right)+F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\
\end{align*}\)
This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 41 rules.
Finding the specification took 1439 seconds.
Copy 41 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_{39}\! \left(x \right) F_{5}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x , y_{0}\right)+F_{6}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , 1, y_{0}\right)\\
F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{29}\! \left(x , y_{1}\right) F_{9}\! \left(x , y_{0}, y_{1}\right)\\
F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}, y_{1}\right)+F_{18}\! \left(x , y_{0}, y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{1}\right) F_{29}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{13}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{13}\! \left(x , y_{1}, y_{2}\right)}{y_{0}-1}\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}\right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{16}\! \left(x , y_{0}\right)\\
F_{16}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right) F_{29}\! \left(x , y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}, y_{1}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{24}\! \left(x , 1, y_{1}\right) y_{1}-F_{24}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{27}\! \left(x , 1, y_{1}\right) y_{1}-F_{27}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{0}, y_{1}\right) F_{29}\! \left(x , y_{1}\right)\\
F_{29}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , y_{0}, y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{32}\! \left(x , 1, y_{1}\right) y_{1}-F_{32}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{33}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{1}\right)\\
F_{34}\! \left(x , y_{0}\right) &= F_{29}\! \left(x , y_{0}\right) F_{35}\! \left(x , y_{0}\right)\\
F_{35}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , 1, y_{0}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0}, y_{1}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{38}\! \left(x , y_{0}, y_{1}\right) y_{1}+F_{38}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{1}\right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= x\\
F_{40}\! \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 "Row Placements Tracked Fusion" and has 35 rules.
Finding the specification took 29 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_{32}\! \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_{33}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{14}\! \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_{15}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \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_{14}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{12}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{12}\! \left(x , y_{1}, y_{2}\right)}{y_{0}-1}\\
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_{30}\! \left(x , y_{0}, y_{1}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, y_{1}\right) F_{14}\! \left(x , y_{0}\right)\\
F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{0}, y_{1}\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_{20}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}\right) F_{18}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{19}\! \left(x , 1, y_{1}\right) y_{1}-F_{19}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{21}\! \left(x , y_{0}, y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{22}\! \left(x , 1, y_{1}\right) y_{1}-F_{22}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{24}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{25}\! \left(x , 1, y_{1}\right) y_{1}-F_{25}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{27}\! \left(x , y_{1}\right)\\
F_{27}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{0}\right)\\
F_{28}\! \left(x , y_{0}\right) &= F_{29}\! \left(x , y_{0}, 1\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , y_{0}, y_{1}\right) F_{32}\! \left(x \right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{12}\! \left(x , y_{0}, y_{1}\right)+F_{12}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{32}\! \left(x \right) &= x\\
F_{33}\! \left(x , y_{0}\right) &= F_{32}\! \left(x \right) F_{34}\! \left(x , y_{0}\right)\\
F_{34}\! \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*}\)