Av(12543, 13542, 14532, 15432, 23541, 24531, 25431)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3129, 17442, 99574, 579108, 3419056, 20440024, 123494294, 752913720, ...
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 40 rules.
Finding the specification took 908 seconds.
Copy 40 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{13}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{10}\! \left(x \right) &= 0\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \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 \right) &= x\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= -\frac{-F_{22}\! \left(x , y\right) y +F_{22}\! \left(x , 1\right)}{-1+y}\\
F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\
F_{27}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= y x\\
F_{30}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\
F_{32}\! \left(x , y\right) &= y F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{35}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{35}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{27}\! \left(x , y\right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 59 rules.
Finding the specification took 236 seconds.
Copy 59 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_{57}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\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) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y_{0}\right)+F_{56}\! \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_{10}\! \left(x , y_{0}\right) y_{0}+F_{10}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{10}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}\right)+F_{13}\! \left(x , y_{0}\right)+F_{14}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{3}\! \left(x \right)\\
F_{12}\! \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_{13}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y_{0}\right)\\
F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}\right) F_{25}\! \left(x , y_{0}\right)\\
F_{15}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , 1, y_{0}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0} y_{1}, y_{1}\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_{44}\! \left(x , y_{0}, y_{1}\right)+F_{52}\! \left(x , y_{0}, y_{1}\right)+F_{53}\! \left(x , y_{1}\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}, y_{1}\right) y_{0}+F_{20}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{24}\! \left(x , y_{0}\right)+F_{51}\! \left(x , y_{1}, y_{0}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{17}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{17}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y_{0}\right) &= F_{25}\! \left(x , y_{0}\right) F_{26}\! \left(x , y_{0}\right)\\
F_{25}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{26}\! \left(x , y_{0}\right) &= F_{27}\! \left(x , 1, y_{0}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{28}\! \left(x , y_{0} y_{1}\right) y_{0}+F_{28}\! \left(x , y_{1}\right)}{-1+y_{0}}\\
F_{28}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y_{0}\right)+F_{48}\! \left(x , y_{0}\right)+F_{50}\! \left(x , y_{0}\right)\\
F_{29}\! \left(x , y_{0}\right) &= F_{30}\! \left(x , 1, y_{0}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{31}\! \left(x , y_{0}, y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y_{0}, y_{1}\right)+F_{43}\! \left(x , y_{0}, y_{1}\right)+F_{44}\! \left(x , y_{0}, y_{1}\right)+F_{47}\! \left(x , y_{0}, y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{34}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , y_{0}, y_{1}\right)+F_{42}\! \left(x , y_{1}, y_{0}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{31}\! \left(x , y_{0}, y_{1}\right) y_{0}+F_{31}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}\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_{41}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}+F_{41}\! \left(x , y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{28}\! \left(x , y_{0}\right)-y_{1} F_{28}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}\right) F_{41}\! \left(x , y_{1}, y_{0}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}\right) F_{45}\! \left(x , y_{0}, y_{1}\right)\\
F_{45}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{46}\! \left(x , y_{0}, 1\right)-y_{1} F_{46}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{46}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{47}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{1}\right) F_{31}\! \left(x , y_{0}, y_{1}\right)\\
F_{48}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{49}\! \left(x , y_{0}\right)\\
F_{49}\! \left(x , y_{0}\right) &= F_{34}\! \left(x , 1, y_{0}\right)\\
F_{50}\! \left(x , y_{0}\right) &= F_{25}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{0}\right)\\
F_{51}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{1}\right) F_{3}\! \left(x \right)\\
F_{53}\! \left(x , y_{0}\right) &= F_{25}\! \left(x , y_{0}\right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{3}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{56}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{3}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{3}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
\end{align*}\)