###### Av(12354, 13254)
Counting Sequence
1, 1, 2, 6, 24, 118, 672, 4258, 29241, 213865, 1645677, 13204357, 109723588, 939262476, 8247972113, ...

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 54 rules.

Found on January 24, 2022.

Finding the specification took 177 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_{21}\! \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_{51}\! \left(x , y_{0}, y_{1}\right)+F_{52}\! \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) &= F_{14}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{0}-F_{14}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{21}\! \left(x , y_{0}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{19}\! \left(x , y_{0}, y_{1}\right) y_{1}-F_{19}\! \left(x , y_{0}, y_{2}\right) y_{2}}{-y_{2}+y_{1}}\\ F_{19}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{20}\! \left(x , y_{0}, 1\right) y_{0}-F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{21}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{1}\right) F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-F_{24}\! \left(x , 1, y_{1}, y_{2}\right) y_{1}+F_{24}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right) y_{0}}{-y_{1}+y_{0}}\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{7}\! \left(x \right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{38}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{41}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) F_{7}\! \left(x \right)\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) y_{0}-F_{28}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right)}{-1+y_{0}}\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{0}\right) F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{33}\! \left(x , y_{0}, y_{1}, y_{3}\right) y_{1}-F_{33}\! \left(x , y_{0}, y_{2}, y_{3}\right) y_{2}}{-y_{2}+y_{1}}\\ F_{33}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{34}\! \left(x , y_{0}, 1, y_{2}\right) y_{0}-F_{34}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{1}\right) F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-F_{37}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right) y_{1}+F_{37}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}, y_{3}\right) y_{0}}{-y_{1}+y_{0}}\\ F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{2}-F_{24}\! \left(x , y_{0}, y_{1}, y_{3}\right) y_{3}}{-y_{3}+y_{2}}\\ F_{38}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{2}\right) F_{39}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{39}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right) y_{2}-F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, \frac{y_{3}}{y_{2}}\right) y_{3}}{-y_{3}+y_{2}}\\ F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{2} y_{3}\right)\\ F_{41}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{21}\! \left(x , y_{3}\right) F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}\right) F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{44}\! \left(x , y_{0}, y_{1}\right) y_{1}-F_{44}\! \left(x , y_{0}, y_{2}\right) y_{2}}{-y_{2}+y_{1}}\\ F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{1}\right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{47}\! \left(x , y_{0}, y_{1}, 1\right) y_{1}-F_{47}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right) y_{2}}{-y_{2}+y_{1}}\\ F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{2}\right) F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{2}\right) F_{50}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{50}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{40}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right)\\ F_{51}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}\right) F_{44}\! \left(x , y_{0}, y_{1}\right)\\ F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{1}\right) F_{53}\! \left(x , y_{0}, y_{1}\right)\\ F_{53}\! \left(x , y_{0}, y_{1}\right) &= F_{47}\! \left(x , y_{0}, y_{1}, 1\right)\\ \end{align*}