###### Av(1243)
Counting Sequence
1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, ...
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 "Row Placements Tracked Fusion" and has 26 rules.

Found on May 28, 2021.

Finding the specification took 5 seconds.

Copy to clipboard:

View tree on standalone page.

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

### This specification was found using the strategy pack "Insertion Col Placements Tracked Fusion Req Corrob" and has 53 rules.

Found on May 28, 2021.

Finding the specification took 41 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 53 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_{6}\! \left(x \right)+F_{9}\! \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_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{20}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , 1, y\right)\\ F_{32}\! \left(x , y , z\right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x , y , z\right)+F_{37}\! \left(x , z , y\right)\\ F_{33}\! \left(x , y , z\right) &= \frac{z \left(F_{34}\! \left(x , y , 1\right)-F_{34}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{34}\! \left(x , y , z\right) &= F_{35}\! \left(x , y , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{36}\! \left(x , y , z\right)\\ F_{36}\! \left(x , y , z\right) &= F_{28}\! \left(x , z\right)+F_{32}\! \left(x , y , z\right)\\ F_{37}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{38}\! \left(x , z , y\right)\\ F_{38}\! \left(x , y , z\right) &= F_{16}\! \left(x , y\right)+F_{32}\! \left(x , y , z\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , 1, y\right)\\ F_{40}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z \right)+F_{41}\! \left(x , y , z\right)\\ F_{41}\! \left(x , y , z\right) &= F_{42}\! \left(x , y , y z \right)\\ F_{42}\! \left(x , y , z\right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x , y , z\right)+F_{46}\! \left(x , y , z\right)+F_{48}\! \left(x , z , y\right)\\ F_{43}\! \left(x , y , z\right) &= -\frac{y \left(F_{44}\! \left(x , 1, z\right)-F_{44}\! \left(x , y , z\right)\right)}{-1+y}\\ F_{44}\! \left(x , y , z\right) &= F_{4}\! \left(x \right) F_{45}\! \left(x , y , z\right)\\ F_{45}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)+F_{42}\! \left(x , y , z\right)\\ F_{46}\! \left(x , y , z\right) &= \frac{z \left(F_{47}\! \left(x , y , 1\right)-F_{47}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{47}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{40}\! \left(x , y , z\right)\\ F_{48}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{49}\! \left(x , z , y\right)\\ F_{49}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right)+F_{42}\! \left(x , y , z\right)\\ F_{50}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{41}\! \left(x , y , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 27 rules.

Found on May 28, 2021.

Finding the specification took 15 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 27 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{10}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ 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 F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , 1, y\right)\\ F_{14}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)\\ F_{15}\! \left(x , y , z\right) &= F_{10}\! \left(x \right) F_{16}\! \left(x , y , z\right)\\ F_{16}\! \left(x , y , z\right) &= -\frac{-y F_{14}\! \left(x , y , z\right)+F_{14}\! \left(x , 1, z\right)}{-1+y}\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{20}\! \left(x , y\right)\\ F_{18}\! \left(x , y , z\right) &= \frac{y F_{19}\! \left(x , y , 1\right)-z F_{19}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{19}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , y z \right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{20}\! \left(x , z\right)\\ F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , 1, y\right)-F_{24}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\ F_{24}\! \left(x , y , z\right) &= F_{14}\! \left(x , y z , z\right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Insertion Row Placements Tracked Fusion Req Corrob" and has 53 rules.

Found on May 28, 2021.

Finding the specification took 39 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 53 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_{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 , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{20}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , 1, y\right)\\ F_{32}\! \left(x , y , z\right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x , y , z\right)+F_{37}\! \left(x , y , z\right)\\ F_{33}\! \left(x , y , z\right) &= \frac{z \left(F_{34}\! \left(x , y , 1\right)-F_{34}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{34}\! \left(x , y , z\right) &= F_{35}\! \left(x , y , y z \right)\\ F_{35}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{36}\! \left(x , y , z\right)\\ F_{36}\! \left(x , y , z\right) &= F_{28}\! \left(x , z\right)+F_{32}\! \left(x , y , z\right)\\ F_{37}\! \left(x , y , z\right) &= F_{20}\! \left(x , z\right) F_{38}\! \left(x , y , z\right)\\ F_{38}\! \left(x , y , z\right) &= F_{16}\! \left(x , y\right)+F_{32}\! \left(x , y , z\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , 1, y\right)\\ F_{40}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z \right)+F_{41}\! \left(x , y , z\right)\\ F_{41}\! \left(x , y , z\right) &= F_{42}\! \left(x , y , y z \right)\\ F_{42}\! \left(x , y , z\right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x , y , z\right)+F_{46}\! \left(x , y , z\right)+F_{48}\! \left(x , y , z\right)\\ F_{43}\! \left(x , y , z\right) &= -\frac{y \left(F_{44}\! \left(x , 1, z\right)-F_{44}\! \left(x , y , z\right)\right)}{-1+y}\\ F_{44}\! \left(x , y , z\right) &= F_{45}\! \left(x , y , z\right) F_{8}\! \left(x \right)\\ F_{45}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)+F_{42}\! \left(x , y , z\right)\\ F_{46}\! \left(x , y , z\right) &= \frac{z \left(F_{47}\! \left(x , y , 1\right)-F_{47}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\ F_{47}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{40}\! \left(x , y , z\right)\\ F_{48}\! \left(x , y , z\right) &= F_{20}\! \left(x , z\right) F_{49}\! \left(x , y , z\right)\\ F_{49}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right)+F_{42}\! \left(x , y , z\right)\\ F_{50}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{41}\! \left(x , y , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 26 rules.

Found on May 28, 2021.

Finding the specification took 5 seconds.

Copy to clipboard:

View tree on standalone page.

Copy 26 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_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \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_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{-y F_{12}\! \left(x , y\right)+F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , 1, y\right)\\ F_{13}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{20}\! \left(x , z , y\right)\\ F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{15}\! \left(x , y , z\right) &= -\frac{-y F_{13}\! \left(x , y , z\right)+F_{13}\! \left(x , 1, z\right)}{-1+y}\\ F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y , z\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y , z\right) &= \frac{y F_{19}\! \left(x , y , 1\right)-z F_{19}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{19}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , y z \right)\\ F_{20}\! \left(x , y , z\right) &= F_{13}\! \left(x , z , y\right) F_{17}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= \frac{y F_{23}\! \left(x , 1, y\right)-F_{23}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\ F_{23}\! \left(x , y , z\right) &= F_{13}\! \left(x , y z , z\right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ \end{align*}