Av(12345, 12354, 12435, 13425, 21345, 21354, 21435, 31245, 31254)
Counting Sequence
1, 1, 2, 6, 24, 111, 551, 2843, 15008, 80361, 434304, 2361950, 12901991, 70699255, 388312757, ...
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 71 rules.
Finding the specification took 167 seconds.
Copy 71 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_{1}\! \left(x \right)+F_{5}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{50}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{8}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= -\frac{-F_{7}\! \left(x , y_{0}\right) y_{0}+F_{7}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right) F_{3}\! \left(x \right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , 1, y_{0}\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_{53}\! \left(x , y_{0}, y_{1}\right)+F_{55}\! \left(x , y_{0}, y_{1}\right)+F_{68}\! \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_{18}\! \left(x , y_{0}, y_{1}\right)+F_{52}\! \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_{3}\! \left(x \right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{11}\! \left(x , y_{0}\right) y_{0}-F_{11}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{21}\! \left(x , y_{0}, 1\right) y_{0}-F_{21}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{50}\! \left(x , y_{0}\right)+F_{52}\! \left(x , y_{1}, y_{0}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{42}\! \left(x , y_{0}, y_{1}\right)+F_{43}\! \left(x , y_{0}, y_{1}\right)+F_{47}\! \left(x , y_{0}\right)+F_{49}\! \left(x , y_{1}, y_{0}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}+F_{28}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
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_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{37}\! \left(x , y_{1}, y_{0}\right)+F_{40}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{3}\! \left(x \right) F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}+F_{28}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{3}\! \left(x \right) F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{17}\! \left(x , y_{0}, y_{2}\right) y_{0}-F_{17}\! \left(x , y_{1}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}\right) F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{35}\! \left(x , y_{0}, 1, y_{2}\right) y_{0}-F_{35}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{36}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{22}\! \left(x , y_{0}, y_{1}\right) y_{1}-F_{22}\! \left(x , y_{0}, y_{2}\right) y_{2}}{-y_{2}+y_{1}}\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{38}\! \left(x , y_{1}, y_{0}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{39}\! \left(x , 1, y_{1}\right) y_{1}-F_{39}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}\right) F_{41}\! \left(x , y_{1}, y_{2}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{7}\! \left(x , y_{0}\right) y_{0}-F_{7}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{44}\! \left(x , y_{0}, y_{1}\right)\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{45}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{45}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{45}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{46}\! \left(x , y_{0}\right) y_{0}-F_{46}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{46}\! \left(x , y_{0}\right) &= F_{22}\! \left(x , 1, y_{0}\right)\\
F_{47}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right) F_{48}\! \left(x , y_{0}\right)\\
F_{48}\! \left(x , y_{0}\right) &= F_{38}\! \left(x , 1, y_{0}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{50}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right) F_{51}\! \left(x , y_{0}\right)\\
F_{51}\! \left(x , y_{0}\right) &= F_{39}\! \left(x , 1, y_{0}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{7}\! \left(x , y_{1}\right)\\
F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{54}\! \left(x , y_{0}, y_{1}\right)\\
F_{54}\! \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_{55}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{56}\! \left(x , y_{0}, y_{1}\right)\\
F_{56}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{57}\! \left(x , y_{0}, 1\right) y_{0}-F_{57}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{57}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{58}\! \left(x , y_{0}, y_{1}\right)+F_{62}\! \left(x , y_{0}\right)+F_{65}\! \left(x , y_{0}\right)+F_{67}\! \left(x , y_{1}, y_{0}\right)\\
F_{58}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{59}\! \left(x , y_{0}, y_{1}\right)\\
F_{59}\! \left(x , y_{0}, y_{1}\right) &= F_{60}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{60}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{F_{61}\! \left(x , 1, y_{1}, y_{2}\right) y_{1}-y_{0} F_{61}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{61}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{62}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{63}\! \left(x , y_{0}\right)\\
F_{63}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{64}\! \left(x , y_{0}\right)+F_{64}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{64}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{62}\! \left(x , y_{0}\right)+F_{65}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{65}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right) F_{66}\! \left(x , y_{0}\right)\\
F_{66}\! \left(x , y_{0}\right) &= F_{57}\! \left(x , y_{0}, 1\right)\\
F_{67}\! \left(x , y_{0}, y_{1}\right) &= F_{68}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{68}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{64}\! \left(x , y_{1}\right)\\
F_{69}\! \left(x \right) &= F_{3}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{64}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 64 rules.
Finding the specification took 288 seconds.
Copy 64 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 \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\
F_{10}\! \left(x \right) &= 0\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= -\frac{y \left(F_{9}\! \left(x , 1\right)-F_{9}\! \left(x , y\right)\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= y F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= -\frac{y \left(F_{17}\! \left(x , 1\right)-F_{17}\! \left(x , y\right)\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= y x\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{5}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= y F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= 3 F_{10}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= -\frac{-y F_{28}\! \left(x , y\right)+F_{28}\! \left(x , 1\right)}{-1+y}\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{32}\! \left(x , y\right)+F_{48}\! \left(x \right)\\
F_{31}\! \left(x , y\right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= -\frac{-F_{30}\! \left(x , y\right) y +F_{30}\! \left(x , 1\right)}{-1+y}\\
F_{36}\! \left(x \right) &= 2 F_{10}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x , y\right) &= -\frac{-y F_{31}\! \left(x , y\right)+F_{31}\! \left(x , 1\right)}{-1+y}\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\
F_{48}\! \left(x \right) &= -F_{5}\! \left(x \right)-F_{54}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{58}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{59}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= y F_{28}\! \left(x , y\right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\
\end{align*}\)