###### Av(1243, 1324, 1432, 34125)
Counting Sequence
1, 1, 2, 6, 21, 78, 298, 1158, 4554, 18073, 72242, 290448, 1173306, 4758477, 19362607, ...
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 Col Placements Tracked Fusion Req Corrob" and has 41 rules.

Found on July 20, 2021.

Finding the specification took 113 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\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_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , 1, y\right)\\ F_{16}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{19}\! \left(x , y , z\right)+F_{33}\! \left(x , z , y\right)+F_{35}\! \left(x , z , y\right)\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x , y , z\right) &= \frac{F_{16}\! \left(x , y , z\right) y -F_{16}\! \left(x , 1, z\right)}{-1+y}\\ F_{19}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{20}\! \left(x , y , z\right)\\ F_{20}\! \left(x , y , z\right) &= F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y , z\right)\\ F_{21}\! \left(x , y , z\right) &= \frac{z \left(F_{22}\! \left(x , y , 1\right)-F_{22}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{22}\! \left(x , y , z\right) &= 2 F_{8}\! \left(x \right)+F_{23}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)\\ F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , z , y\right)\\ F_{25}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{26}\! \left(x , y , z\right)\\ F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y z , z\right)\\ F_{27}\! \left(x , y , z\right) &= \frac{F_{28}\! \left(x , y , z\right) z -F_{28}\! \left(x , y , 1\right)}{-1+z}\\ F_{29}\! \left(x , y , z\right) &= F_{28}\! \left(x , y z , z\right)\\ F_{30}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{29}\! \left(x , y , z\right)\\ F_{30}\! \left(x , y , z\right) &= F_{22}\! \left(x , z , y\right)\\ F_{31}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{32}\! \left(x , y , z\right)\\ F_{20}\! \left(x , y , z\right) &= \frac{F_{32}\! \left(x , y , 1\right) y -F_{32}\! \left(x , y , \frac{z}{y}\right) z}{-z +y}\\ F_{33}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{34}\! \left(x , z , y\right)\\ F_{32}\! \left(x , y , z\right) &= F_{34}\! \left(x , y , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{36}\! \left(x , z , y\right)\\ F_{36}\! \left(x , y , z\right) &= \frac{-F_{37}\! \left(x , 1, z\right) z +F_{37}\! \left(x , \frac{y}{z}, z\right) y}{-z +y}\\ F_{37}\! \left(x , y , z\right) &= F_{34}\! \left(x , y z , z\right)\\ F_{38}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ \end{align*}

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

Found on July 20, 2021.

Finding the specification took 41 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_{10}\! \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_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right)+F_{7}\! \left(x \right)+F_{8}\! \left(x , y_{0}\right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x \right) F_{9}\! \left(x , y_{0}\right)\\ F_{9}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}\right)\\ 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 , y_{0}, y_{0} y_{1}\right)\\ F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x \right) F_{16}\! \left(x , y_{0}, y_{1}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{0}, y_{1}\right)+F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x \right) F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, y_{1}\right) F_{22}\! \left(x , y_{0}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{13}\! \left(x , y_{0}, 1\right)-y_{1} F_{13}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{22}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{22}\! \left(x , y_{1}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \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_{14}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{14}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{26}\! \left(x , y_{0}\right) &= F_{22}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}\right)\\ F_{12}\! \left(x , y_{0}\right) &= F_{27}\! \left(x , y_{0}\right)+F_{28}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= F_{6}\! \left(x , y_{0}\right)\\ \end{align*}

### This specification was found using the strategy pack "Point Placements Tracked Fusion Req Corrob" and has 126 rules.

Found on July 20, 2021.

Finding the specification took 412 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_{26}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{26}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{37}\! \left(x , y\right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{24}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= x\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{22}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{26}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{32}\! \left(x \right) &= -F_{36}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= -F_{28}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , 1, y\right)\\ F_{42}\! \left(x , y , z\right) &= F_{15}\! \left(x , z\right) F_{41}\! \left(x , y , z\right)\\ F_{42}\! \left(x , y , z\right) &= F_{43}\! \left(x , y , z\right)\\ F_{43}\! \left(x , y , z\right) &= F_{116}\! \left(x , y , z\right)+F_{44}\! \left(x , z\right)\\ F_{45}\! \left(x , y\right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{49}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{49}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x \right) F_{64}\! \left(x , y\right)\\ F_{61}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{26}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{50}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= \frac{y F_{75}\! \left(x , 1, y\right)-F_{75}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\ F_{41}\! \left(x , y , z\right) &= F_{75}\! \left(x , y , z\right)+F_{76}\! \left(x , y , z\right)\\ F_{77}\! \left(x , y , z\right) &= F_{15}\! \left(x , z\right) F_{76}\! \left(x , y , z\right)\\ F_{77}\! \left(x , y , z\right) &= F_{78}\! \left(x , y , z\right)\\ F_{43}\! \left(x , y , z\right) &= F_{78}\! \left(x , y , z\right)+F_{79}\! \left(x , y , z\right)\\ F_{80}\! \left(x , y , z\right) &= F_{12}\! \left(x , y z \right)+F_{79}\! \left(x , y , z\right)\\ F_{80}\! \left(x , y , z\right) &= F_{112}\! \left(x , y , z\right)+F_{81}\! \left(x , z\right)\\ F_{45}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{84}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= -\frac{y \left(F_{90}\! \left(x , 1\right)-F_{90}\! \left(x , y\right)\right)}{-1+y}\\ F_{90}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , 1, y\right)\\ F_{92}\! \left(x , y , z\right) &= F_{93}\! \left(x , y , z\right)\\ F_{93}\! \left(x , y , z\right) &= F_{26}\! \left(x \right) F_{94}\! \left(x , y , z\right)\\ F_{94}\! \left(x , y , z\right) &= \frac{y z \left(F_{95}\! \left(x , y , z\right)-F_{95}\! \left(x , \frac{1}{z}, z\right)\right)}{y z -1}\\ F_{95}\! \left(x , y , z\right) &= F_{96}\! \left(x , y , z\right)\\ F_{96}\! \left(x , y , z\right) &= F_{15}\! \left(x , z\right) F_{97}\! \left(x , y , z\right)\\ F_{98}\! \left(x , y , z\right) &= F_{15}\! \left(x , z\right) F_{97}\! \left(x , y , z\right)\\ F_{98}\! \left(x , y , z\right) &= F_{99}\! \left(x , y , z\right)\\ F_{99}\! \left(x , y , z\right) &= F_{100}\! \left(x , y , z\right)+F_{102}\! \left(x , y , z\right)\\ F_{100}\! \left(x , y , z\right) &= F_{101}\! \left(x , y , z\right)\\ F_{101}\! \left(x , y , z\right) &= F_{0}\! \left(x \right) F_{49}\! \left(x , z\right) F_{50}\! \left(x , z\right)\\ F_{102}\! \left(x , y , z\right) &= F_{103}\! \left(x , y , z\right)\\ F_{103}\! \left(x , y , z\right) &= F_{104}\! \left(x , y , z\right) F_{26}\! \left(x \right)\\ F_{104}\! \left(x , y , z\right) &= F_{105}\! \left(x , y , z\right)+F_{106}\! \left(x , y , z\right)\\ F_{105}\! \left(x , y , z\right) &= \frac{y z F_{99}\! \left(x , y , z\right)-F_{99}\! \left(x , \frac{1}{z}, z\right)}{y z -1}\\ F_{106}\! \left(x , y , z\right) &= F_{107}\! \left(x , y , z\right) F_{50}\! \left(x , z\right)\\ F_{107}\! \left(x , y , z\right) &= F_{105}\! \left(x , y , z\right)+F_{108}\! \left(x , y z \right)\\ F_{109}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{95}\! \left(x , y , z\right) &= F_{112}\! \left(x , y , z\right)+F_{92}\! \left(x , y , z\right)\\ F_{113}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{50}\! \left(x , y\right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= F_{112}\! \left(x , 1, y\right)\\ F_{95}\! \left(x , y , z\right) &= F_{116}\! \left(x , y , z\right)+F_{37}\! \left(x , y z \right)\\ F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , 1, y\right)\\ F_{120}\! \left(x , y , z\right) &= F_{121}\! \left(x , y , z\right)\\ F_{121}\! \left(x , y , z\right) &= F_{122}\! \left(x , y , z\right) F_{26}\! \left(x \right) F_{49}\! \left(x , z\right)\\ F_{122}\! \left(x , y , z\right) &= \frac{y z \left(F_{123}\! \left(x , y , z\right)-F_{123}\! \left(x , \frac{1}{z}, z\right)\right)}{y z -1}\\ F_{123}\! \left(x , y , z\right) &= F_{120}\! \left(x , y , z\right)+F_{124}\! \left(x , y , z\right)\\ F_{124}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{49}\! \left(x , z\right)\\ F_{125}\! \left(x \right) &= F_{117}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 41 rules.

Found on July 20, 2021.

Finding the specification took 120 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_{10}\! \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_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{7}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , 1, y\right)\\ F_{15}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)+F_{33}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\ F_{16}\! \left(x , y , z\right) &= F_{10}\! \left(x \right) F_{17}\! \left(x , y , z\right)\\ F_{17}\! \left(x , y , z\right) &= -\frac{-y F_{15}\! \left(x , y , z\right)+F_{15}\! \left(x , 1, z\right)}{-1+y}\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right) F_{30}\! \left(x , y\right)\\ F_{19}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right)+F_{20}\! \left(x , y , z\right)\\ F_{20}\! \left(x , y , z\right) &= \frac{z \left(F_{21}\! \left(x , y , 1\right)-F_{21}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{21}\! \left(x , y , z\right) &= 2 F_{7}\! \left(x \right)+F_{22}\! \left(x , y , z\right)+F_{31}\! \left(x , y , z\right)\\ F_{22}\! \left(x , y , z\right) &= F_{10}\! \left(x \right) F_{23}\! \left(x , y , z\right)\\ F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\ F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y , y z \right)\\ F_{26}\! \left(x , y , z\right) &= -\frac{-y F_{27}\! \left(x , y , z\right)+F_{27}\! \left(x , 1, z\right)}{-1+y}\\ F_{28}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , y z \right)\\ F_{29}\! \left(x , y , z\right) &= F_{28}\! \left(x , y , z\right) F_{30}\! \left(x , z\right)\\ F_{29}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right)\\ F_{30}\! \left(x , y\right) &= y x\\ F_{31}\! \left(x , y , z\right) &= F_{30}\! \left(x , z\right) F_{32}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= \frac{y F_{32}\! \left(x , y , 1\right)-z F_{32}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{33}\! \left(x , y , z\right) &= F_{30}\! \left(x , z\right) F_{34}\! \left(x , y , z\right)\\ F_{32}\! \left(x , y , z\right) &= F_{34}\! \left(x , y , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{30}\! \left(x , z\right) F_{36}\! \left(x , y , z\right)\\ F_{36}\! \left(x , y , z\right) &= -\frac{z F_{37}\! \left(x , 1, z\right)-y F_{37}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\ F_{37}\! \left(x , y , z\right) &= F_{34}\! \left(x , y z , z\right)\\ F_{38}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\ \end{align*}

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

Found on July 20, 2021.

Finding the specification took 44 seconds.

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Copy 29 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_{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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y_{0}\right) &= F_{11}\! \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_{4}\! \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 , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\ F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}, 1\right)\\ F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{0}, y_{1}\right)+F_{23}\! \left(x , y_{1}, y_{0}\right)+F_{26}\! \left(x , y_{0}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \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}\right)+F_{23}\! \left(x , y_{1}, y_{0}\right)+F_{24}\! \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_{4}\! \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}\right) &= F_{12}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}, y_{1}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{14}\! \left(x , y_{0}, 1\right)-y_{1} F_{14}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{15}\! \left(x , y_{1}, y_{0}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{25}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, y_{2}\right)-y_{1} F_{15}\! \left(x , y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{26}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}\right)\\ F_{13}\! \left(x , y_{0}\right) &= F_{27}\! \left(x , y_{0}\right)+F_{28}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}\right)\\ \end{align*}