Av(13524, 14523, 15423, 23514, 24513, 25413, 34512, 35412, 45312)
Counting Sequence
1, 1, 2, 6, 24, 111, 546, 2760, 14154, 73265, 381921, 2002585, 10554288, 55880817, 297110895, ...
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 39 rules.
Found on January 23, 2022.Finding the specification took 49 seconds.
Copy 39 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\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= -\frac{-y F_{25}\! \left(x , y\right)+F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26} \left(x \right)^{2} F_{3}\! \left(x \right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+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_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right) F_{31}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{3}\! \left(x \right) F_{31}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 30 rules.
Found on January 23, 2022.Finding the specification took 205 seconds.
Copy 30 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_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \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 , y\right) F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= y x\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)^{2} F_{10}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \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) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= \frac{F_{26}\! \left(x , 1, y\right)-F_{26}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y z , z\right)\\
F_{27}\! \left(x , y , z\right) &= F_{28}\! \left(x , y , z\right)\\
F_{28}\! \left(x , y , z\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{29}\! \left(x , y , z\right) F_{7}\! \left(x , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , z\right)+F_{7}\! \left(x , z\right)\\
\end{align*}\)