Av(12435, 12453, 12543, 14235, 14253, 14325, 41235, 41253, 41325)
Counting Sequence
1, 1, 2, 6, 24, 111, 546, 2769, 14342, 75607, 404702, 2194468, 12029226, 66544651, 370995194, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 60 rules.
Found on January 23, 2022.Finding the specification took 54 seconds.
Copy 60 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_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{6}\! \left(x , y_{0}\right) F_{7}\! \left(x \right)\\
F_{6}\! \left(x , y_{0}\right) &= \frac{y_{0} F_{4}\! \left(x , y_{0}\right)-F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , 1, y_{0}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y_{0}, y_{1}\right)+F_{27}\! \left(x , y_{0}, y_{1}\right)+F_{58}\! \left(x , y_{0}, y_{1}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{1}\right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{55}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\
F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{14}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{21}\! \left(x , y_{0}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{19}\! \left(x , y_{0}, y_{1}\right)-y_{2} F_{19}\! \left(x , y_{0}, y_{2}\right)}{-y_{2}+y_{1}}\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{20}\! \left(x , y_{0}, 1\right)-y_{1} F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{21}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{1}\right) F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-y_{1} F_{24}\! \left(x , 1, y_{1}, y_{2}\right)+y_{0} F_{24}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{53}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{1}\right) F_{31}\! \left(x , y_{1}, y_{2}\right) F_{4}\! \left(x , y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}\right) F_{3}\! \left(x \right) F_{31}\! \left(x , y_{0}, y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y_{0}, y_{1}\right)+F_{36}\! \left(x , y_{0}, y_{1}\right)+F_{39}\! \left(x , y_{0}, y_{1}\right)+F_{42}\! \left(x , y_{0}, y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{35}\! \left(x , y_{0}\right)-y_{1} F_{35}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{35}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , 1, y_{0}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0}, y_{1}\right) F_{7}\! \left(x \right)\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{38}\! \left(x , y_{0}\right)-y_{1} F_{38}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{38}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , 1, y_{0}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{40}\! \left(x , y_{0}, y_{1}\right) F_{7}\! \left(x \right)\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{41}\! \left(x , y_{0}\right)-y_{1} F_{41}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{41}\! \left(x , y_{0}\right) &= F_{28}\! \left(x , 1, y_{0}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{43}\! \left(x , y_{0}, y_{1}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , 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) &= F_{45}\! \left(x , y_{1}\right)+F_{52}\! \left(x , y_{0}, y_{1}\right)\\
F_{45}\! \left(x , y_{0}\right) &= F_{46}\! \left(x , y_{0}\right)+F_{51}\! \left(x , y_{0}\right)\\
F_{46}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x , y_{0}\right)\\
F_{47}\! \left(x , y_{0}\right) &= F_{48}\! \left(x , y_{0}\right)\\
F_{48}\! \left(x , y_{0}\right) &= F_{49}\! \left(x , y_{0}\right)^{2} F_{21}\! \left(x , y_{0}\right)\\
F_{49}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x , y_{0}\right)\\
F_{50}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right) F_{49}\! \left(x , y_{0}\right)\\
F_{51}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right) F_{49}\! \left(x , y_{0}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}\right) F_{49}\! \left(x , y_{1}\right)\\
F_{53}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{54}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{54}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{2}\right) F_{4}\! \left(x , y_{1}\right) F_{44}\! \left(x , y_{1}, y_{2}\right)\\
F_{55}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{2}\right) F_{45}\! \left(x , y_{2}\right) F_{57}\! \left(x , y_{0}, y_{1}\right)\\
F_{57}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{4}\! \left(x , y_{0}\right)-y_{1} F_{4}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{58}\! \left(x , y_{0}, y_{1}\right) &= F_{59}\! \left(x , y_{0}, y_{1}\right)\\
F_{59}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{1}\right) F_{4}\! \left(x , y_{0}\right) F_{45}\! \left(x , y_{1}\right)\\
\end{align*}\)