Av(1324, 1432)
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Counting Sequence
1, 1, 2, 6, 22, 89, 380, 1677, 7566, 34676, 160808, 752608, 3548325, 16830544, 80234659, ...
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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 "All The Strategies 1 Tracked Fusion" and has 42 rules.

Found on December 02, 2021.

Finding the specification took 2094 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 \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{37}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{17}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x , y\right) &= \frac{F_{20}\! \left(x , y\right) y -F_{20}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , 1, y\right)\\ F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right) F_{29}\! \left(x , y\right)\\ F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y\right)+F_{31}\! \left(x , y , z\right)+F_{34}\! \left(x , y , z\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= y x\\ F_{30}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{31}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)\\ F_{32}\! \left(x , y , z\right) &= F_{33}\! \left(x , y , z\right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x , y , z\right) &= \frac{F_{23}\! \left(x , y , z\right) y -F_{23}\! \left(x , 1, z\right)}{-1+y}\\ F_{34}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right) F_{29}\! \left(x , z\right)\\ F_{35}\! \left(x , y , z\right) &= F_{36}\! \left(x , y , z\right)\\ F_{36}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= \frac{F_{16}\! \left(x , y\right) y -F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{39}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Expand Verified" and has 40 rules.

Found on October 17, 2021.

Finding the specification took 39896 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{13}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y_{0}\right) &= F_{17}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\ F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{15}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)\\ F_{16}\! \left(x , y_{0}\right) &= F_{10}\! \left(x \right) F_{14}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\ F_{17}\! \left(x , y_{0}\right) &= F_{18}\! \left(x , y_{0}\right)\\ F_{18}\! \left(x , y_{0}\right) &= F_{13}\! \left(x \right) F_{19}\! \left(x , y_{0}\right)\\ F_{19}\! \left(x , y_{0}\right) &= F_{20}\! \left(x , y_{0}\right)+F_{33}\! \left(x , y_{0}\right)+F_{35}\! \left(x , y_{0}\right)+F_{4}\! \left(x \right)\\ F_{20}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right)\\ F_{21}\! \left(x , y_{0}\right) &= F_{13}\! \left(x \right) F_{22}\! \left(x , y_{0}\right)\\ F_{22}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{23}\! \left(x , y_{0}\right)+F_{23}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{23}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , 1, y_{0}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{31}\! \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_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{8}\! \left(x , y_{0} y_{1}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x \right) F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} y_{1} F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{25}\! \left(x , \frac{1}{y_{1}}, y_{1}, y_{2}\right)}{y_{0} y_{1}-1}\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{2}\right) F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{33}\! \left(x , y_{0}\right) &= F_{13}\! \left(x \right) F_{34}\! \left(x , y_{0}\right)\\ F_{34}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{19}\! \left(x , y_{0}\right)+F_{19}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{35}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{19}\! \left(x , y_{0}\right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x , 1\right)\\ F_{38}\! \left(x , y_{0}\right) &= F_{39}\! \left(x , y_{0}\right)\\ F_{39}\! \left(x , y_{0}\right) &= F_{13}\! \left(x \right) F_{23}\! \left(x , y_{0}\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 50 rules.

Found on October 16, 2021.

Finding the specification took 1980 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 \right) F_{46}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{46}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right)+F_{15}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}\right) &= F_{0}\! \left(x \right) F_{11}\! \left(x , y_{0}\right)\\ F_{11}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y_{0}\right)\\ F_{12}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right)\\ F_{13}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{15}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)\\ F_{16}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right) F_{17}\! \left(x , y_{0}\right) F_{46}\! \left(x \right)\\ F_{17}\! \left(x , y_{0}\right) &= F_{18}\! \left(x , y_{0}\right)+F_{19}\! \left(x , y_{0}\right)\\ F_{18}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{19}\! \left(x , y_{0}\right) &= F_{20}\! \left(x , y_{0}\right)\\ F_{20}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right) F_{46}\! \left(x \right)\\ F_{21}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{22}\! \left(x , y_{0}\right)+F_{22}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{22}\! \left(x , y_{0}\right) &= F_{23}\! \left(x , 1, y_{0}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{26}\! \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_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{27}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{11}\! \left(x , y_{1}\right) F_{11}\! \left(x , y_{3}\right) F_{9}\! \left(x , y_{3}\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_{11}\! \left(x , y_{0}\right) F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{46}\! \left(x \right)\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{32}\! \left(x , y_{0} y_{1}, y_{2}\right)+F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{1}\right) F_{34}\! \left(x , y_{0}, y_{1}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{35}\! \left(x , 1, y_{0}, y_{1}\right)-F_{35}\! \left(x , \frac{1}{y_{0}}, y_{0}, y_{1}\right)}{-1+y_{0}}\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} \left(F_{37}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)-F_{37}\! \left(x , \frac{1}{y_{0}}, y_{0}, y_{1}, y_{2}\right)\right)}{-1+y_{0}}\\ F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{38}\! \left(x , y_{0} y_{1}, y_{1} y_{2}, y_{3}\right)\\ F_{38}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{39}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{40}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0}\right) F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{43}\! \left(x \right)\\ F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{43} \left(x \right)^{2} F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= x\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{46}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 92 rules.

Found on May 25, 2021.

Finding the specification took 18951 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 \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{8}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{8}\! \left(x \right) &= 0\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= -\frac{y \left(F_{7}\! \left(x , 1\right)-F_{7}\! \left(x , y\right)\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , 1, y\right)\\ F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y z , z\right)\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right)+F_{6}\! \left(x , z\right)\\ F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right)\\ F_{17}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x , y , z\right)\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y\right)+F_{29}\! \left(x , y , z\right)\\ F_{20}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , 1, y\right)\\ F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z , z\right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)+F_{27}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)+F_{8}\! \left(x \right)\\ F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y , z\right) &= -\frac{-y F_{24}\! \left(x , y , z\right)+F_{24}\! \left(x , 1, z\right)}{-1+y}\\ F_{27}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{24}\! \left(x , y , z\right)\\ F_{28}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x , z , y\right)\\ F_{29}\! \left(x , y , z\right) &= F_{30}\! \left(x , z , y\right)\\ F_{30}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{31}\! \left(x , z , y\right)\\ F_{32}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{31}\! \left(x , y , z\right)\\ F_{32}\! \left(x , y , z\right) &= F_{33}\! \left(x , y , z\right)\\ F_{34}\! \left(x , y , z\right) &= F_{33}\! \left(x , y , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{34}\! \left(x , y , z\right)+F_{91}\! \left(x , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{36}\! \left(x , y , z\right)+F_{80}\! \left(x , y z \right)\\ F_{36}\! \left(x , y , z\right) &= F_{37}\! \left(x , y , y z \right)\\ F_{37}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right)+F_{38}\! \left(x , y , z\right)\\ F_{38}\! \left(x , y , z\right) &= F_{39}\! \left(x , y , z\right)\\ F_{39}\! \left(x , y , z\right) &= F_{4}\! \left(x \right) F_{40}\! \left(x , y , z\right)\\ F_{40}\! \left(x , y , z\right) &= -\frac{-y F_{41}\! \left(x , y , z\right)+F_{41}\! \left(x , 1, z\right)}{-1+y}\\ F_{41}\! \left(x , y , z\right) &= F_{42}\! \left(x , z\right)+F_{74}\! \left(x , y , z\right)\\ F_{43}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{49}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{54}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{61}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{65}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , 1, y\right)\\ F_{69}\! \left(x , y , z\right) &= F_{16}\! \left(x , y z , z\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= \frac{y \left(F_{73}\! \left(x , 1, y\right)-F_{73}\! \left(x , \frac{1}{y}, y\right)\right)}{-1+y}\\ F_{73}\! \left(x , y , z\right) &= F_{74}\! \left(x , y z , z\right)\\ F_{74}\! \left(x , y , z\right) &= F_{75}\! \left(x , y , z\right)\\ F_{75}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{31}\! \left(x , y , z\right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{76} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{79}\! \left(x , y\right) &= F_{73}\! \left(x , 1, y\right)\\ F_{81}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{80}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{16}\! \left(x , y , 1\right)\\ F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= -\frac{y \left(F_{88}\! \left(x , 1\right)-F_{88}\! \left(x , y\right)\right)}{-1+y}\\ F_{88}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Tracked Fusion Expand Verified" and has 161 rules.

Found on October 17, 2021.

Finding the specification took 39415 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 \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{15}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{15}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{0}\right)\\ F_{16}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\ F_{19}\! \left(x , y_{0}\right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x , y_{0}\right)\\ F_{20}\! \left(x , y_{0}\right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x , y_{0}\right)+F_{30}\! \left(x , y_{0}\right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x , y_{0}\right) &= F_{23}\! \left(x , y_{0}\right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{24}\! \left(x , 1\right)-F_{24}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{24}\! \left(x , y_{0}\right) &= F_{20}\! \left(x , y_{0}\right)+F_{25}\! \left(x , y_{0}\right)\\ F_{25}\! \left(x , y_{0}\right) &= F_{26}\! \left(x , y_{0}\right)\\ F_{26}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}\right)\\ F_{27}\! \left(x , y_{0}\right) &= F_{28}\! \left(x , y_{0}\right)+F_{29}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}\right)\\ F_{29}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}\right) F_{16}\! \left(x , y_{0}\right)\\ F_{30}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{31}\! \left(x , y_{0}\right)\\ F_{31}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right)+F_{32}\! \left(x , y_{0}\right)\\ F_{32}\! \left(x , y_{0}\right) &= F_{33}\! \left(x , y_{0}, 1\right)\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{138}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{35}\! \left(x , y_{0} y_{1}\right)\\ F_{35}\! \left(x , y_{0}\right) &= F_{159}\! \left(x , y_{0}\right)+F_{21}\! \left(x \right)+F_{36}\! \left(x , y_{0}\right)\\ F_{36}\! \left(x , y_{0}\right) &= F_{37}\! \left(x , y_{0}\right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x , y_{0}\right) &= F_{38}\! \left(x , y_{0}\right)+F_{49}\! \left(x , y_{0}\right)\\ F_{38}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{39}\! \left(x , y_{0}\right)\\ F_{39}\! \left(x , y_{0}\right) &= F_{21}\! \left(x \right)+F_{40}\! \left(x , y_{0}\right)+F_{44}\! \left(x , y_{0}\right)\\ F_{40}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{41}\! \left(x , y_{0}\right)\\ F_{41}\! \left(x , y_{0}\right) &= F_{38}\! \left(x , y_{0}\right)+F_{42}\! \left(x , y_{0}\right)\\ F_{42}\! \left(x , y_{0}\right) &= F_{43}\! \left(x , y_{0}\right)+F_{46}\! \left(x , y_{0}\right)\\ F_{43}\! \left(x , y_{0}\right) &= F_{44}\! \left(x , y_{0}\right)\\ F_{44}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{45}\! \left(x , y_{0}\right)\\ F_{45}\! \left(x , y_{0}\right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x , y_{0}\right)\\ F_{46}\! \left(x , y_{0}\right) &= F_{47}\! \left(x , y_{0}\right)\\ F_{47}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{48}\! \left(x , y_{0}\right)\\ F_{48}\! \left(x , y_{0}\right) &= F_{42}\! \left(x , y_{0}\right)\\ F_{49}\! \left(x , y_{0}\right) &= F_{50}\! \left(x , 1, y_{0}\right)\\ F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{1}\right)+F_{51}\! \left(x , y_{0}, y_{1}\right)\\ F_{51}\! \left(x , y_{0}, y_{1}\right) &= F_{157}\! \left(x , y_{1}, y_{0}\right)+F_{21}\! \left(x \right)+F_{52}\! \left(x , y_{0}, y_{1}\right)+F_{66}\! \left(x , y_{0}, y_{1}\right)\\ F_{52}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{0} \left(F_{53}\! \left(x , 1, y_{1}\right)-F_{53}\! \left(x , y_{0}, y_{1}\right)\right)}{-1+y_{0}}\\ F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{4}\! \left(x \right) F_{54}\! \left(x , y_{0}, y_{1}\right)\\ F_{54}\! \left(x , y_{0}, y_{1}\right) &= F_{51}\! \left(x , y_{0}, y_{1}\right)+F_{55}\! \left(x , y_{0}, y_{1}\right)\\ F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{55}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{57}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{57}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{2}\right) F_{15}\! \left(x , y_{2}\right) F_{58}\! \left(x , y_{1}\right)\\ F_{58}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{59}\! \left(x , y_{0}\right)\\ F_{59}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{60}\! \left(x , y_{0}\right)\\ F_{60}\! \left(x , y_{0}\right) &= F_{21}\! \left(x \right)+F_{61}\! \left(x , y_{0}\right)+F_{63}\! \left(x , y_{0}\right)\\ F_{61}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{62}\! \left(x , y_{0}\right)\\ F_{62}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{60}\! \left(x , y_{0}\right)\\ F_{63}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{64}\! \left(x , y_{0}\right)\\ F_{64}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{65}\! \left(x , y_{0}\right)\\ F_{65}\! \left(x , y_{0}\right) &= F_{63}\! \left(x , y_{0}\right)\\ F_{66}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}\right) F_{67}\! \left(x , y_{0}, y_{1}\right)\\ F_{67}\! \left(x , y_{0}, y_{1}\right) &= F_{50}\! \left(x , y_{0}, y_{1}\right)+F_{68}\! \left(x , y_{0}, y_{1}\right)\\ F_{68}\! \left(x , y_{0}, y_{1}\right) &= F_{69}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{69}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{70}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{70}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{150}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{71}\! \left(x , y_{0} y_{1}, y_{3}\right)\\ F_{71}\! \left(x , y_{0}, y_{1}\right) &= F_{72}\! \left(x , y_{1}, y_{0}\right)\\ F_{73}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{72}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\ F_{73}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{1}\right) F_{74}\! \left(x , y_{2}, y_{0}, y_{1}\right)\\ F_{74}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{75}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{75}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{76}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{83}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{76}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{77}\! \left(x , y_{2}, y_{1}, y_{0}\right)\\ F_{77}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{78}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{79}\! \left(x , y_{0}, y_{1}\right) &= F_{78}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{79}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{1}\right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{83}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{114}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{84}\! \left(x , y_{0} y_{1}\right)\\ F_{84}\! \left(x , y_{0}\right) &= F_{85}\! \left(x , y_{0}\right)\\ F_{85}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{86}\! \left(x , y_{0}\right)\\ F_{86}\! \left(x , y_{0}\right) &= F_{87}\! \left(x , y_{0}\right)+F_{97}\! \left(x , y_{0}\right)\\ F_{87}\! \left(x , y_{0}\right) &= F_{88}\! \left(x , y_{0}\right)+F_{93}\! \left(x , y_{0}\right)\\ F_{88}\! \left(x , y_{0}\right) &= F_{89}\! \left(x , y_{0}\right)+F_{92}\! \left(x , y_{0}\right)\\ F_{89}\! \left(x , y_{0}\right) &= F_{90}\! \left(x , y_{0}\right)\\ F_{90}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{91}\! \left(x , y_{0}\right)\\ F_{91}\! \left(x , y_{0}\right) &= F_{16}\! \left(x , y_{0}\right)+F_{89}\! \left(x , y_{0}\right)\\ F_{92}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{35}\! \left(x , 1\right)-F_{35}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{93}\! \left(x , y_{0}\right) &= F_{94}\! \left(x , y_{0}\right)+F_{96}\! \left(x , y_{0}\right)\\ F_{94}\! \left(x , y_{0}\right) &= F_{95}\! \left(x , y_{0}\right)\\ F_{95}\! \left(x , y_{0}\right) &= F_{11}\! \left(x \right) F_{16}\! \left(x , y_{0}\right) F_{80}\! \left(x \right)\\ F_{96}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{84}\! \left(x , 1\right)-F_{84}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{97}\! \left(x , y_{0}\right) &= F_{103}\! \left(x , y_{0}\right)+F_{98}\! \left(x , y_{0}\right)\\ F_{98}\! \left(x , y_{0}\right) &= F_{94}\! \left(x , y_{0}\right)+F_{99}\! \left(x , y_{0}\right)\\ F_{99}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{100}\! \left(x , 1\right)-F_{100}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{100}\! \left(x , y_{0}\right) &= F_{101}\! \left(x , y_{0}\right)\\ F_{101}\! \left(x , y_{0}\right) &= F_{102}\! \left(x , y_{0}\right) F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{102}\! \left(x , y_{0}\right) &= F_{100}\! \left(x , y_{0}\right)+F_{35}\! \left(x , y_{0}\right)\\ F_{103}\! \left(x , y_{0}\right) &= F_{104}\! \left(x , y_{0}\right)+F_{110}\! \left(x , y_{0}\right)\\ F_{104}\! \left(x , y_{0}\right) &= F_{105}\! \left(x , y_{0}\right)\\ F_{105}\! \left(x , y_{0}\right) &= F_{106}\! \left(x , 1, y_{0}\right)\\ F_{106}\! \left(x , y_{0}, y_{1}\right) &= F_{107}\! \left(x , y_{0}\right) F_{11}\! \left(x \right) F_{16}\! \left(x , y_{1}\right) F_{80}\! \left(x \right)\\ F_{107}\! \left(x , y_{0}\right) &= F_{108}\! \left(x , y_{0}\right)\\ F_{108}\! \left(x , y_{0}\right) &= F_{109}\! \left(x , y_{0}\right)^{2} F_{14}\! \left(x , y_{0}\right)\\ F_{109}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{107}\! \left(x , y_{0}\right)\\ F_{110}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{111}\! \left(x , 1\right)-F_{111}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{111}\! \left(x , y_{0}\right) &= F_{112}\! \left(x , y_{0}\right)\\ F_{112}\! \left(x , y_{0}\right) &= F_{113}\! \left(x , y_{0}\right) F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{113}\! \left(x , y_{0}\right) &= F_{111}\! \left(x , y_{0}\right)+F_{84}\! \left(x , y_{0}\right)\\ F_{114}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{115}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{115}\! \left(x , y_{0}, y_{1}\right) &= F_{116}\! \left(x , y_{0}, y_{1}\right)\\ F_{116}\! \left(x , y_{0}, y_{1}\right) &= F_{117}\! \left(x , y_{0}, y_{1}\right)\\ F_{117}\! \left(x , y_{0}, y_{1}\right) &= F_{118}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x , y_{0}, y_{1}\right) &= F_{119}\! \left(x , y_{0}, y_{1}\right)+F_{130}\! \left(x , y_{0}, y_{1}\right)\\ F_{119}\! \left(x , y_{0}, y_{1}\right) &= F_{120}\! \left(x , y_{0}, y_{1}\right)+F_{125}\! \left(x , y_{0}, y_{1}\right)\\ F_{120}\! \left(x , y_{0}, y_{1}\right) &= F_{121}\! \left(x , y_{0}, y_{1}\right)+F_{124}\! \left(x , y_{0}, y_{1}\right)\\ F_{122}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{121}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{122}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{123}\! \left(x , y_{2}, y_{1}, y_{0}\right)\\ F_{123}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x \right) F_{16}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{2}\right)\\ F_{124}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{0} \left(F_{71}\! \left(x , 1, y_{1}\right)-F_{71}\! \left(x , y_{0}, y_{1}\right)\right)}{-1+y_{0}}\\ F_{125}\! \left(x , y_{0}, y_{1}\right) &= F_{126}\! \left(x , y_{0}, y_{1}\right)+F_{129}\! \left(x , y_{0}, y_{1}\right)\\ F_{127}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{126}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{127}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{128}\! \left(x , y_{2}, y_{1}, y_{0}\right)\\ F_{128}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x \right) F_{16}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{2}\right) F_{80}\! \left(x \right)\\ F_{129}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{0} \left(F_{116}\! \left(x , 1, y_{1}\right)-F_{116}\! \left(x , y_{0}, y_{1}\right)\right)}{-1+y_{0}}\\ F_{130}\! \left(x , y_{0}, y_{1}\right) &= F_{131}\! \left(x , y_{0}, y_{1}\right)+F_{139}\! \left(x , y_{0}, y_{1}\right)\\ F_{131}\! \left(x , y_{0}, y_{1}\right) &= F_{126}\! \left(x , y_{0}, y_{1}\right)+F_{132}\! \left(x , y_{0}, y_{1}\right)\\ F_{132}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{0} \left(F_{133}\! \left(x , 1, y_{1}\right)-F_{133}\! \left(x , y_{0}, y_{1}\right)\right)}{-1+y_{0}}\\ F_{133}\! \left(x , y_{0}, y_{1}\right) &= F_{134}\! \left(x , y_{0}, y_{1}\right)\\ F_{135}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{134}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{135}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{136}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{136}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{137}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{137}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{135}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{138}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{138}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{73}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\ F_{139}\! \left(x , y_{0}, y_{1}\right) &= F_{140}\! \left(x , y_{0}, y_{1}\right)+F_{146}\! \left(x , y_{0}, y_{1}\right)\\ F_{140}\! \left(x , y_{0}, y_{1}\right) &= F_{141}\! \left(x , y_{1}, y_{0}\right)\\ F_{141}\! \left(x , y_{0}, y_{1}\right) &= F_{142}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{142}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{143}\! \left(x , y_{1}, y_{0}, y_{2}\right)\\ F_{144}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{143}\! \left(x , y_{0}, y_{1}, y_{0} y_{2}\right)\\ F_{145}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{144}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{145}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{2}\right)^{2} F_{80} \left(x \right)^{2} F_{16}\! \left(x , y_{0}\right)\\ F_{146}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{0} \left(F_{147}\! \left(x , 1, y_{1}\right)-F_{147}\! \left(x , y_{0}, y_{1}\right)\right)}{-1+y_{0}}\\ F_{147}\! \left(x , y_{0}, y_{1}\right) &= F_{148}\! \left(x , y_{0}, y_{1}\right)\\ F_{148}\! \left(x , y_{0}, y_{1}\right) &= F_{149}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{149}\! \left(x , y_{0}, y_{1}\right) &= F_{116}\! \left(x , y_{0}, y_{1}\right)+F_{147}\! \left(x , y_{0}, y_{1}\right)\\ F_{150}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{151}\! \left(x , y_{1}, y_{2}, y_{0}, y_{3}\right)\\ F_{151}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{14}\! \left(x , y_{1}\right) F_{152}\! \left(x , y_{2}, y_{0}, y_{1}, y_{3}\right)\\ F_{152}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{y_{3} \left(F_{153}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{153}\! \left(x , y_{0}, y_{1}, \frac{y_{3}}{y_{1}}\right)\right)}{y_{1} y_{2}-y_{3}}\\ F_{153}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{138}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{154}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{154}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{114}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{155}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{155}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{156}\! \left(x , y_{2}, y_{1}, y_{0}\right)\\ F_{156}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{2}\right) F_{80}\! \left(x \right)\\ F_{157}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}\right) F_{158}\! \left(x , y_{1}, y_{0}\right)\\ F_{158}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{0}\right)+F_{71}\! \left(x , y_{0}, y_{1}\right)\\ F_{159}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{160}\! \left(x , y_{0}\right)\\ F_{160}\! \left(x , y_{0}\right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x , y_{0}\right)\\ \end{align*}\)