Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14325, 23145, 23154, 23415, 23514, 24135, 24315, 34125, 34215)
Counting Sequence
1, 1, 2, 6, 24, 100, 426, 1875, 8482, 39143, 183476, 871216, 4182214, 20263013, 98958258, ...
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 50 rules.
Found on January 23, 2022.Finding the specification took 104 seconds.
Copy 50 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{31}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{28}\! \left(x , y\right) &= y x\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= -\frac{-y F_{25}\! \left(x , y\right)+F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= -\frac{-y F_{33}\! \left(x , y\right)+F_{33}\! \left(x , 1\right)}{-1+y}\\
F_{33}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{36}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\
F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{31}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x , 1\right)\\
F_{41}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x , 1\right)\\
F_{43}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 24 rules.
Found on January 22, 2022.Finding the specification took 112 seconds.
Copy 24 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_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\
F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{21}\! \left(x , z , y\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , z , y\right)\\
F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y\right) F_{3}\! \left(x \right) F_{7}\! \left(x , z\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , z , y\right)\\
F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y z , y\right)\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\
F_{18}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right)+F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right)\\
F_{20}\! \left(x , y , z\right) &= F_{14}\! \left(x , z\right) F_{5}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , y\right) F_{5}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
\end{align*}\)