Av(12345, 12354, 13245, 13254)
Counting Sequence
1, 1, 2, 6, 24, 116, 632, 3728, 23268, 151484, 1018824, 7030256, 49522764, 354794596, 2577806856, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 54 rules.
Found on January 24, 2022.Finding the specification took 245 seconds.
Copy 54 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_{51}\! \left(x , y_{0}, y_{1}\right)+F_{52}\! \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_{49}\! \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_{25}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{38}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{41}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) F_{7}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{y_{0} F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)-F_{28}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right)}{-1+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{0}\right) F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{y_{1} F_{33}\! \left(x , y_{0}, y_{1}, y_{3}\right)-y_{2} F_{33}\! \left(x , y_{0}, y_{2}, y_{3}\right)}{-y_{2}+y_{1}}\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{34}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{34}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{1}\right) F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-y_{1} F_{37}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right)+y_{0} F_{37}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}, y_{3}\right)}{-y_{1}+y_{0}}\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{y_{2} F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)-y_{3} F_{24}\! \left(x , y_{0}, y_{1}, y_{3}\right)}{-y_{3}+y_{2}}\\
F_{38}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{2}\right) F_{39}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{y_{2} F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right)-y_{3} F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, \frac{y_{3}}{y_{2}}\right)}{-y_{3}+y_{2}}\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{2} y_{3}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{21}\! \left(x , y_{3}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}\right) F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{44}\! \left(x , y_{0}, y_{1}\right)-y_{2} F_{44}\! \left(x , y_{0}, y_{2}\right)}{-y_{2}+y_{1}}\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{1}\right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{47}\! \left(x , y_{0}, y_{1}, 1\right)-y_{2} F_{47}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right)}{-y_{2}+y_{1}}\\
F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0}, y_{1}\right) F_{21}\! \left(x , y_{2}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{2}\right) F_{50}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{50}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right)\\
F_{51}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}\right) F_{44}\! \left(x , y_{0}, y_{1}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{1}\right) F_{53}\! \left(x , y_{0}, y_{1}\right)\\
F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{47}\! \left(x , y_{0}, y_{1}, 1\right)\\
\end{align*}\)