###### Av(1234, 1324)
Counting Sequence
1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074, 176351644, ...
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 And Col Placements Tracked Fusion Expand Verified" and has 25 rules.

Found on November 11, 2021.

Finding the specification took 339 seconds.

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\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_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y_{0}\right) &= F_{23}\! \left(x \right) F_{5}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \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 , y_{0}, y_{0} y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{9}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{1}, y_{0}\right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{6}\! \left(x , y_{1}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{1}, y_{0}\right) F_{12}\! \left(x , y_{1}\right)\\ F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, 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_{23}\! \left(x \right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{1}, y_{0}, y_{2}\right)+F_{17}\! \left(x , y_{1}, y_{2}\right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{1}\right) F_{15}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{7}\! \left(x , y_{0}, 1\right)-y_{1} F_{7}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{19}\! \left(x , y_{0} y_{1}, y_{2}\right) y_{0}-F_{19}\! \left(x , y_{1}, y_{2}\right)}{y_{0}-1}\\ F_{19}\! \left(x , y_{0}, y_{1}\right) &= \frac{-y_{1} F_{20}\! \left(x , 1, y_{1}\right)+y_{0} F_{20}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{2} F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{14}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x , y_{0}\right) &= \frac{y_{0} F_{4}\! \left(x , y_{0}\right)-F_{4}\! \left(x , 1\right)}{y_{0}-1}\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 31 rules.

Found on April 20, 2021.

Finding the specification took 1147 seconds.

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\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_{3}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{6}\! \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_{6}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)+F_{7}\! \left(x , y_{0}, y_{1}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}\right) F_{8}\! \left(x , y_{0}, y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}\right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{7}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}\right) F_{8}\! \left(x , y_{0}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{1}\right) F_{12}\! \left(x , y_{0}, y_{1}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{13}\! \left(x , 1, y_{1}\right)-y_{0} F_{13}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, 1\right)-y_{1} F_{15}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{1}\right) F_{28}\! \left(x \right)\\ F_{17}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{18}\! \left(x , y_{0}, 1\right)-y_{1} F_{18}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{23}\! \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_{7}\! \left(x , y_{1}, y_{2}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{2}\right) F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{12}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{12}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{28}\! \left(x \right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{28}\! \left(x \right) &= x\\ F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x \right) F_{30}\! \left(x , y_{0}, y_{1}\right)\\ F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{6}\! \left(x , y_{0}, y_{1}\right)+F_{6}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\ \end{align*}

### This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 26 rules.

Found on November 11, 2021.

Finding the specification took 294 seconds.

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\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_{24}\! \left(x \right) F_{4}\! \left(x , y_{0}\right)\\ F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{6}\! \left(x , y_{0}, 1\right)\\ F_{6}\! \left(x , y_{0}, y_{1}\right) &= F_{7}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}\right) F_{8}\! \left(x , y_{0}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{1}, y_{0}\right)+F_{12}\! \left(x , y_{0}, y_{1}\right)+F_{5}\! \left(x , y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{1}\right) F_{9}\! \left(x , y_{1}, y_{0}\right)\\ F_{11}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x \right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{1}, y_{0}, y_{2}\right)+F_{16}\! \left(x , y_{1}, y_{2}\right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{1}\right) F_{14}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{6}\! \left(x , y_{0}, 1\right)-y_{1} F_{6}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} y_{1} F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right)+y_{2} F_{20}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{2}\right)}{y_{0} y_{1}-y_{2}}\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{1} F_{22}\! \left(x , y_{0} y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{2}\right)}{y_{1}-1}\\ F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{1}\right) F_{8}\! \left(x , y_{0}, y_{1}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{13}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{24}\! \left(x \right) &= x\\ F_{25}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{3}\! \left(x , y_{0}\right)+F_{3}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 29 rules.

Found on April 20, 2021.

Finding the specification took 72 seconds.

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\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_{26}\! \left(x \right) F_{3}\! \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_{27}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , 1, y_{0}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{5}\! \left(x , 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_{1}\right) F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{10}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{26}\! \left(x \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 , y_{0}, y_{1}, 1\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_{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_{18}\! \left(x , y_{1}, y_{2}\right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{0}\right) F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}\right) F_{19}\! \left(x , y_{0}, y_{1}\right)\\ 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_{8}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{2}\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_{0} F_{23}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{23}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{7}\! \left(x , 1, y_{1}\right)-y_{0} F_{7}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\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_{2}\right) F_{26}\! \left(x \right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{16}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{26}\! \left(x \right) &= x\\ F_{27}\! \left(x , y_{0}\right) &= F_{26}\! \left(x \right) F_{28}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{4}\! \left(x , y_{0}\right)+F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 29 rules.

Found on April 23, 2021.

Finding the specification took 57 seconds.

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\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_{26}\! \left(x \right) F_{3}\! \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_{27}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , 1, y_{0}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{5}\! \left(x , y_{1} y_{2}\right)+F_{9}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{1}\right) F_{8}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{10}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{26}\! \left(x \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 , y_{0}, y_{1}, 1\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_{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_{18}\! \left(x , y_{1}, y_{2}\right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{0}\right) F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}\right) F_{19}\! \left(x , y_{0}, y_{1}\right)\\ 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_{8}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{2}\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_{0} F_{23}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{23}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{7}\! \left(x , 1, y_{1}\right)-y_{0} F_{7}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\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_{2}\right) F_{26}\! \left(x \right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{16}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{26}\! \left(x \right) &= x\\ F_{27}\! \left(x , y_{0}\right) &= F_{26}\! \left(x \right) F_{28}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{4}\! \left(x , y_{0}\right)+F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}