Av(12345, 12453, 13452, 23451)
Counting Sequence
1, 1, 2, 6, 24, 116, 633, 3747, 23502, 153928, 1042595, 7253909, 51590639, 373708913, 2749481916, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 45 rules.
Found on January 24, 2022.Finding the specification took 295 seconds.
Copy 45 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_{43}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{37}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y , 1\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , y z \right)\\
F_{13}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{19}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x , y , z\right) &= \frac{F_{13}\! \left(x , y , z\right) y -F_{13}\! \left(x , 1, z\right)}{-1+y}\\
F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right) F_{18}\! \left(x , y\right)\\
F_{17}\! \left(x , y , z\right) &= \frac{F_{12}\! \left(x , y , 1\right) y -F_{12}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{18}\! \left(x , y\right) &= y x\\
F_{19}\! \left(x , y , z\right) &= F_{18}\! \left(x , z\right) F_{20}\! \left(x , y , z\right)\\
F_{20}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{22}\! \left(x , y , z\right) &= \frac{F_{20}\! \left(x , y , z\right) y -F_{20}\! \left(x , 1, z\right)}{-1+y}\\
F_{23}\! \left(x , y , z\right) &= F_{18}\! \left(x , y\right) F_{24}\! \left(x , y , z\right)\\
F_{24}\! \left(x , y , z\right) &= \frac{F_{25}\! \left(x , y , 1\right) y -F_{25}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{25}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , y z \right)\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x , y , z\right) &= \frac{F_{28}\! \left(x , y , z\right) z -F_{28}\! \left(x , y , 1\right)}{-1+z}\\
F_{28}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)+F_{34}\! \left(x , y , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{30}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{30}\! \left(x , y , z\right) &= \frac{F_{28}\! \left(x , y , z\right) y -F_{28}\! \left(x , 1, z\right)}{-1+y}\\
F_{31}\! \left(x , y , z\right) &= F_{18}\! \left(x , y\right) F_{32}\! \left(x , y , z\right)\\
F_{32}\! \left(x , y , z\right) &= \frac{F_{33}\! \left(x , y , 1\right) y -F_{33}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\
F_{33}\! \left(x , y , z\right) &= F_{28}\! \left(x , y , y z \right)\\
F_{34}\! \left(x , y , z\right) &= F_{18}\! \left(x , z\right) F_{28}\! \left(x , y , z\right)\\
F_{35}\! \left(x , y , z\right) &= F_{36}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{36}\! \left(x , y , z\right) &= \frac{F_{20}\! \left(x , y , z\right) z -F_{20}\! \left(x , y , 1\right)}{-1+z}\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{38}\! \left(x , y\right) &= \frac{F_{39}\! \left(x , y\right) y -F_{39}\! \left(x , 1\right)}{-1+y}\\
F_{39}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{40}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{41}\! \left(x , y\right) &= \frac{F_{39}\! \left(x , y\right) y -F_{39}\! \left(x , 1\right)}{-1+y}\\
F_{42}\! \left(x , y\right) &= F_{19}\! \left(x , y , 1\right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\
\end{align*}\)