###### Av(12345, 12354)
Counting Sequence
1, 1, 2, 6, 24, 118, 672, 4256, 29176, 212586, 1625704, 12930160, 106242392, 897210996, 7756325952, ...

### This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified" and has 23 rules.

Found on January 22, 2022.

Finding the specification took 30 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_{17}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{17}\! \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_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , 1, y\right)\\ F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y z , z\right)\\ F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{13}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\ F_{11}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{14}\! \left(x , y , z\right)\\ F_{14}\! \left(x , y , z\right) &= \frac{-z F_{9}\! \left(x , 1, z\right)+y F_{9}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{17}\! \left(x \right)\\ F_{16}\! \left(x , y , z\right) &= \frac{z F_{10}\! \left(x , y , z\right)-F_{10}\! \left(x , y , 1\right)}{-1+z}\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{17}\! \left(x \right)\\ F_{19}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Expand Verified" and has 32 rules.

Found on January 22, 2022.

Finding the specification took 42 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 , 1\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}\right) F_{8}\! \left(x \right)\\ F_{7}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{5}\! \left(x , y_{0}\right)+F_{5}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{18}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , 1, y_{0}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{15}\! \left(x , y_{0}, y_{1}\right)+F_{19}\! \left(x , y_{0}, y_{1}\right)\\ F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{8}\! \left(x \right)\\ F_{14}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{12}\! \left(x , y_{0}, y_{1}\right)+F_{12}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{1}\right) F_{18}\! \left(x , y_{0}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{17}\! \left(x , y_{0}, 1\right)-y_{1} F_{17}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{18}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{1}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1} y_{2}, 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_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{8}\! \left(x \right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}\right) F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{27}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{27}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{1}\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_{1} F_{30}\! \left(x , y_{0}, y_{1}, 1\right)-y_{2} F_{30}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right)}{-y_{2}+y_{1}}\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1}\right) F_{18}\! \left(x , y_{2}\right)\\ \end{align*}

### This specification was found using the strategy pack "Row Placements Tracked Fusion Req Corrob Expand Verified" and has 31 rules.

Found on January 22, 2022.

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