###### Av(12453, 12534, 12543, 13452, 13524, 13542, 14523, 14532, 23451, 23514, 23541, 24513, 24531, 34512, 34521)
Counting Sequence
1, 1, 2, 6, 24, 105, 477, 2218, 10504, 50516, 246167, 1213329, 6039842, 30326676, 153431887, ...

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 53 rules.

Found on January 23, 2022.

Finding the specification took 27 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{45}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{33}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)+F_{31}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x \right)+F_{28}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{28}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= y x\\ F_{30}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{16}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{35}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\ F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{33}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\ F_{40}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{36}\! \left(x , 1\right)\\ F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{41}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ \end{align*}

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 27 rules.

Found on January 22, 2022.

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