###### Av(12345, 12354, 12435, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14325, 14352, 41235, 41325, 41352)
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 "Row Placements Tracked Fusion" and has 35 rules.

Found on January 23, 2022.

Finding the specification took 7 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\ F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{29}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{25}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= y x\\ F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{22}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{31}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\ \end{align*}