Av(1243, 1432)
Counting Sequence
1, 1, 2, 6, 22, 89, 382, 1711, 7922, 37663, 182936, 904302, 4535994, 23034564, 118209806, ...
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Expand Verified" and has 40 rules.
Found on November 09, 2021.Finding the specification took 108 seconds.
Copy 40 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 , 1\right)\\
F_{3}\! \left(x , y_{0}\right) &= F_{17}\! \left(x \right) F_{4}\! \left(x , y_{0}\right)\\
F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y_{0}\right)+F_{6}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= \frac{F_{3}\! \left(x , y_{0}\right) y_{0}-F_{3}\! \left(x , 1\right)}{-1+y_{0}}\\
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_{18}\! \left(x , 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_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{17}\! \left(x \right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}\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_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{28}\! \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_{17}\! \left(x \right)\\
F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}-F_{14}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\
F_{18}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{20}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}-F_{20}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{22}\! \left(x , y_{0}, y_{1}, 1\right) y_{0} y_{1}-F_{22}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{0} y_{1}}\right) y_{2}}{y_{0} y_{1}-y_{2}}\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{1}\right) F_{25}\! \left(x , y_{0}, y_{2}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{9}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-F_{29}\! \left(x , y_{1}, y_{2}\right) y_{1} y_{2}+F_{29}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}\right) y_{0}}{-y_{1} y_{2}+y_{0}}\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{31}\! \left(x , y_{0} y_{1}, 1\right) y_{0}-F_{31}\! \left(x , y_{0} y_{1}, \frac{y_{2}}{y_{0}}\right) y_{2}}{-y_{2}+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}\right) F_{33}\! \left(x , y_{1}\right)\\
F_{33}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y_{0}\right)+F_{39}\! \left(x , y_{0}\right)\\
F_{34}\! \left(x , y_{0}\right) &= F_{35}\! \left(x , 1, y_{0}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x \right) F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y_{0}, y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , y_{0}, y_{1}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{35}\! \left(x , y_{0}, y_{1}\right) y_{0}-F_{35}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{39}\! \left(x , y_{0}\right) &= F_{31}\! \left(x , y_{0}, 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified" and has 41 rules.
Found on November 09, 2021.Finding the specification took 154 seconds.
Copy 41 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 , 1\right)\\
F_{3}\! \left(x , y_{0}\right) &= F_{19}\! \left(x \right) F_{4}\! \left(x , y_{0}\right)\\
F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y_{0}\right)+F_{6}\! \left(x , y_{0}\right)\\
F_{5}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{3}\! \left(x , y_{0}\right)+F_{3}\! \left(x , 1\right)}{-1+y_{0}}\\
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 , y_{0}, 1, y_{1}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{9}\! \left(x , y_{0}, 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_{2}\right) F_{39}\! \left(x , y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\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}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{19}\! \left(x \right)\\
F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{16}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{21}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{1}+F_{21}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\
F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} y_{1} F_{23}\! \left(x , y_{0}, y_{1}, 1\right)-y_{2} F_{23}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{0} y_{1}}\right)}{y_{0} y_{1}-y_{2}}\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} y_{2} F_{27}\! \left(x , y_{1}, y_{2}\right)-y_{0} F_{27}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}\right)}{-y_{1} y_{2}+y_{0}}\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{31}\! \left(x , y_{0} y_{1}, 1\right)-y_{2} F_{31}\! \left(x , y_{0} y_{1}, \frac{y_{2}}{y_{0}}\right)}{-y_{2}+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{1}\right) F_{39}\! \left(x , y_{0}\right)\\
F_{33}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y_{0}\right)+F_{38}\! \left(x , y_{0}\right)\\
F_{34}\! \left(x , y_{0}\right) &= F_{35}\! \left(x , 1, y_{0}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x \right) F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{38}\! \left(x , y_{0}\right) &= F_{31}\! \left(x , y_{0}, 1\right)\\
F_{39}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 60 rules.
Found on April 25, 2021.Finding the specification took 791 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right)+F_{8}\! \left(x \right)+F_{9}\! \left(x , y_{0}\right)\\
F_{8}\! \left(x \right) &= 0\\
F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{11}\! \left(x \right)\\
F_{10}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{7}\! \left(x , 1\right)-F_{7}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}, 1\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_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{22}\! \left(x , y_{2}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0} y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y_{0}\right)\\
F_{20}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right)\\
F_{21}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}\right)\\
F_{22}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x , y_{0}\right)+F_{24}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x \right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}\right) F_{26}\! \left(x , y_{0}, y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x , y_{0} y_{1}\right)+F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{1}\right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{48}\! \left(x , y_{0}, y_{1}\right)+F_{59}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{31}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x \right) F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{33}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{36}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x \right) F_{38}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{36}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}\right) F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{41}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{41}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{41}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{17}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{0} y_{1}-F_{17}\! \left(x , y_{0}, \frac{y_{2}}{y_{0}}\right)}{y_{0} y_{1}-1}\\
F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{1}\right) F_{44}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{44}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{2}\right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{46}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{15}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{15}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{48}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}\right) F_{49}\! \left(x , y_{0}, y_{1}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{50}\! \left(x , y_{0} y_{1}\right)+F_{50}\! \left(x , y_{0}\right)}{y_{1}-1}\\
F_{50}\! \left(x , y_{0}\right) &= F_{51}\! \left(x , y_{0}\right)\\
F_{51}\! \left(x , y_{0}\right) &= F_{11}\! \left(x \right) F_{19}\! \left(x , y_{0}\right) F_{52}\! \left(x , y_{0}\right)\\
F_{52}\! \left(x , y_{0}\right) &= F_{53}\! \left(x , 1, y_{0}\right)\\
F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{54}\! \left(x , y_{0}, y_{1}\right)+F_{56}\! \left(x , y_{0}, y_{1}\right)+F_{58}\! \left(x , y_{0}, y_{1}\right)\\
F_{54}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x \right) F_{55}\! \left(x , y_{0}, y_{1}\right)\\
F_{55}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{53}\! \left(x , y_{0}, y_{1}\right)+F_{53}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{56}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}\right) F_{57}\! \left(x , y_{0}, y_{1}\right)\\
F_{57}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{17}\! \left(x , y_{0}, 1\right)-y_{1} F_{17}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{58}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{59}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{1}\right) F_{29}\! \left(x , y_{0}, y_{1}\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 54 rules.
Found on April 25, 2021.Finding the specification took 829 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_{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_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{8}\! \left(x , 1, y_{0}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{5}\! \left(x , y_{0}\right)-y_{1} F_{5}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}, 1\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{1}\right) F_{13}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{1}, y_{2}, y_{0}\right)+F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{27}\! \left(x , y_{1}, y_{0}, y_{2}\right)+F_{29}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{19}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\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_{0} F_{24}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{24}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{26}\! \left(x , y_{0}, y_{1} y_{2}\right) y_{0} y_{1}-F_{26}\! \left(x , y_{0}, \frac{y_{2}}{y_{0}}\right)}{y_{0} y_{1}-1}\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{30}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{30}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{13}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{13}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\
F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{1}\right) F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{34}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{36}\! \left(x , y_{0}, y_{1}\right)+F_{44}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0} y_{1}\right)+F_{40}\! \left(x , y_{0}, y_{1}\right)\\
F_{37}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x , y_{0}\right)\\
F_{38}\! \left(x , y_{0}\right) &= F_{39}\! \left(x , y_{0}\right)\\
F_{39}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{37}\! \left(x , y_{0}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= F_{38}\! \left(x , y_{0}\right)+F_{41}\! \left(x , y_{0}, y_{1}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= F_{42}\! \left(x , y_{0}, y_{1}\right)\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{43}\! \left(x , y_{0}, y_{1}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{38}\! \left(x , y_{0} y_{1}\right)+F_{41}\! \left(x , y_{0}, y_{1}\right)\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{45}\! \left(x , y_{0} y_{1}\right)+F_{45}\! \left(x , y_{0}\right)}{y_{1}-1}\\
F_{45}\! \left(x , y_{0}\right) &= F_{46}\! \left(x , y_{0}\right)\\
F_{46}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{37}\! \left(x , y_{0}\right) F_{47}\! \left(x , y_{0}\right)\\
F_{47}\! \left(x , y_{0}\right) &= F_{48}\! \left(x , 1, y_{0}\right)\\
F_{48}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{49}\! \left(x , y_{0}, y_{1}\right)+F_{51}\! \left(x , y_{0}, y_{1}\right)+F_{53}\! \left(x , y_{1}, y_{0}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{50}\! \left(x , y_{0}, y_{1}\right)\\
F_{50}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{48}\! \left(x , y_{0}, y_{1}\right)+F_{48}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{51}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{52}\! \left(x , y_{0}, y_{1}\right)\\
F_{52}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{26}\! \left(x , y_{0}, 1\right) y_{0}-F_{26}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion Expand Verified" and has 44 rules.
Found on November 07, 2021.Finding the specification took 110 seconds.
Copy 44 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_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{8}\! \left(x , 1, y_{0}\right)\\
F_{8}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{5}\! \left(x , y_{0}\right)-y_{1} F_{5}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}, 1\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{1}\right) F_{13}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{1}, y_{2}, y_{0}\right)+F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{27}\! \left(x , y_{1}, y_{2}, y_{0}\right)+F_{30}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{18}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-y_{1} y_{2} F_{23}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)+y_{0} F_{23}\! \left(x , y_{0}, \frac{1}{y_{2}}, y_{2}\right)}{-y_{1} y_{2}+y_{0}}\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} F_{13}\! \left(x , y_{0}, y_{1}, 1\right)-y_{2} F_{13}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right)}{-y_{2}+y_{1}}\\
F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0} y_{1}, y_{2}, y_{1}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{26}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{31}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} y_{2} F_{32}\! \left(x , y_{1}, y_{2}\right)-y_{0} F_{32}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}\right)}{-y_{1} y_{2}+y_{0}}\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{1}\right) F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{35}\! \left(x , y_{0} y_{1}\right) y_{0}-y_{2} F_{35}\! \left(x , y_{1} y_{2}\right)}{y_{0}-y_{2}}\\
F_{35}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y_{0}\right)+F_{43}\! \left(x , y_{0}\right)\\
F_{36}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{37}\! \left(x , y_{0}\right)\\
F_{37}\! \left(x , y_{0}\right) &= F_{38}\! \left(x , 1, y_{0}\right)\\
F_{38}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x , y_{0}, y_{1}\right)+F_{41}\! \left(x , y_{0}, y_{1}\right)+F_{42}\! \left(x , y_{1}, y_{0}\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{40}\! \left(x , y_{0}, y_{1}\right)\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{38}\! \left(x , y_{0}, y_{1}\right)+F_{38}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= F_{29}\! \left(x , y_{0}, 1, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{42}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\
F_{43}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{35}\! \left(x , y_{0}\right)\\
\end{align*}\)