Av(123456)
Counting Sequence
1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 55 rules.
Found on January 29, 2022.Finding the specification took 312 seconds.
Copy 55 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_{38}\! \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_{53}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}, 1\right)\\
F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{51}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{9}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{9}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{12}\! \left(x , y_{0}, y_{1} y_{2}, y_{2} y_{3}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{2}, 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_{39}\! \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_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{1}\right)\\
F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{17}\! \left(x , y_{0}, y_{1}, y_{2}, y_{2} y_{3}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{19}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{20}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}, y_{3}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{27}\! \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_{33}\! \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_{21}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{22}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}, y_{3}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{0}\right) F_{25}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{24}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{25}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{26}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}, y_{3}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{1}\right) F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-F_{19}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right) y_{1}+F_{19}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}, y_{3}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{3}\right) F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-F_{31}\! \left(x , y_{0}, 1, y_{2}, y_{3}\right) y_{2} y_{3}+F_{31}\! \left(x , y_{0}, \frac{y_{1}}{y_{2} y_{3}}, y_{2}, y_{3}\right) y_{1}}{-y_{2} y_{3}+y_{1}}\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{32}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{20}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{3}\right) F_{34}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) y_{2}-F_{20}\! \left(x , y_{0}, y_{1}, 1, y_{3}\right)}{-1+y_{2}}\\
F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) F_{38}\! \left(x \right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, 1\right) y_{3}-F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, \frac{1}{y_{3}}\right)}{-1+y_{3}}\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, y_{4}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3} y_{4}\right)\\
F_{38}\! \left(x \right) &= x\\
F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{41}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}, y_{1}\right) y_{0} y_{1}-F_{41}\! \left(x , \frac{1}{y_{1}}, y_{2}, y_{1}\right) y_{2}}{y_{0} y_{1}-y_{2}}\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0} y_{1}, y_{2}, y_{1}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{1}\right) F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{44}\! \left(x , y_{0}, y_{1}, 1\right) y_{1}-F_{44}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right) y_{2}}{y_{1}-y_{2}}\\
F_{44}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x \right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right) y_{1}-F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, \frac{1}{y_{1}}\right)}{-1+y_{1}}\\
F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{2}, y_{1} y_{3}\right)\\
F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-F_{50}\! \left(x , 1, y_{1} y_{2}\right) y_{2}+F_{50}\! \left(x , \frac{y_{0}}{y_{2}}, y_{1} y_{2}\right) y_{0}}{-y_{2}+y_{0}}\\
F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{51}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x \right) F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{1} y_{2}-F_{8}\! \left(x , y_{0}, y_{1}, \frac{1}{y_{1}}\right)}{y_{1} y_{2}-1}\\
F_{53}\! \left(x , y_{0}\right) &= F_{38}\! \left(x \right) F_{54}\! \left(x , y_{0}\right)\\
F_{54}\! \left(x , y_{0}\right) &= \frac{F_{4}\! \left(x , y_{0}\right) y_{0}-F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\
\end{align*}\)