###### Av(12354, 12453, 13254, 13452, 14253, 14352, 15243, 15342, 21354, 21453, 23451, 24351, 25341, 31452, 32451)
Counting Sequence
1, 1, 2, 6, 24, 105, 479, 2247, 10778, 52650, 261099, 1311203, 6654836, 34082534, 175919360, ...

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

Found on January 23, 2022.

Finding the specification took 9 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_{23}\! \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) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ \end{align*}

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 18 rules.

Found on January 22, 2022.

Finding the specification took 7 seconds.

Copy to clipboard:

View tree on standalone page.

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