###### Av(12453, 12534, 12543, 13524, 21453, 21534, 21543, 23514, 31452, 31524, 31542, 32514, 41523, 41532, 42513)
Counting Sequence
1, 1, 2, 6, 24, 105, 477, 2224, 10574, 51030, 249180, 1228412, 6104177, 30538331, 153673943, ...

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 40 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \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_{31}\! \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) &= F_{12}\! \left(x , 1, y\right)\\ F_{12}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)+F_{22}\! \left(x , z , y\right)\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{14}\! \left(x , y , z\right) &= \frac{y F_{15}\! \left(x , y , z\right)-F_{15}\! \left(x , 1, z\right)}{-1+y}\\ F_{15}\! \left(x , y , z\right) &= \frac{y F_{7}\! \left(x , y\right)-z F_{7}\! \left(x , z\right)}{-z +y}\\ F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{17}\! \left(x , y , z\right) &= \frac{y F_{12}\! \left(x , y , z\right)-F_{12}\! \left(x , 1, z\right)}{-1+y}\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right) F_{20}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)\\ F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right)\\ F_{23}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right) F_{24}\! \left(x , z , y\right) F_{27}\! \left(x , y\right)\\ F_{24}\! \left(x , y , z\right) &= F_{20}\! \left(x , y\right)+F_{25}\! \left(x , z\right)\\ F_{25}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{25}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{33}\! \left(x \right) &= F_{3}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x , 1\right)\\ F_{35}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= \frac{y F_{35}\! \left(x , y\right)-F_{35}\! \left(x , 1\right)}{-1+y}\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{35}\! \left(x , y\right)\\ \end{align*}