###### Av(12543)
Counting Sequence
1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, ...

### This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified Sliding" and has 35 rules.

Found on January 22, 2022.

Finding the specification took 11 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{F_{4}\! \left(x , y_{0}\right) y_{0}-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{F_{11}\! \left(x , y_{0}, y_{1}\right) y_{0}-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{F_{16}\! \left(x , y_{0}, 1\right) y_{0}-F_{16}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-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}, y_{1}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{27}\! \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_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}-F_{22}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{29}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{30}\! \left(x , y_{0}, 1, y_{2}\right) y_{0}-F_{30}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , y_{1}\right) F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , y_{2}\right) F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-F_{34}\! \left(x , y_{0}, 1, y_{2}\right) y_{2}+F_{34}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right) y_{1}}{-y_{2}+y_{1}}\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Req Corrob Expand Verified Sliding" and has 36 rules.

Found on January 22, 2022.

Finding the specification took 12 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}, y_{1}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{33}\! \left(x , y_{0}, 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_{8}\! \left(x \right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{23}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}\right) F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{30}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{31}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{31}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{1}\right) F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{2}\right) F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{2} F_{35}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{35}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right)}{-y_{2}+y_{1}}\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{2}, y_{1}\right)\\ \end{align*}