###### Av(13452, 13524, 13542, 14352, 31452, 31524, 31542, 34152, 34512, 35124, 35142, 35412, 41352, 43152, 43512)
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 76 rules.

Found on January 23, 2022.

Finding the specification took 49 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_{32}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{32}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)+F_{70}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{62}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{32}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{20}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x , y\right)+F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)^{2} F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{21}\! \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_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{27}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{32}\! \left(x \right) &= x\\ F_{33}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= -\frac{-y F_{35}\! \left(x , y\right)+F_{35}\! \left(x , 1\right)}{-1+y}\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{27}\! \left(x , y\right) F_{3}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{38}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{42}\! \left(x \right) F_{43}\! \left(x , y\right)\\ F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{44}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{6}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{53}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{55}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{27}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{61}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= -\frac{-y F_{45}\! \left(x , y\right)+F_{45}\! \left(x , 1\right)}{-1+y}\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x , y\right)\\ F_{65}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{18}\! \left(x \right) F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{70}\! \left(x , y\right) &= F_{32}\! \left(x \right) F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{43}\! \left(x , y\right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{32}\! \left(x \right) F_{65}\! \left(x \right)\\ \end{align*}

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 17 rules.

Found on January 22, 2022.

Finding the specification took 4 seconds.

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Copy 17 equations to clipboard:
\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\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= \frac{y F_{10}\! \left(x , 1, y\right)-F_{10}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\ F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{15}\! \left(x , y , z\right)\\ F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right)\\ F_{12}\! \left(x , y , z\right) &= F_{10}\! \left(x , y , z\right) F_{13}\! \left(x , z\right) F_{14}\! \left(x , z\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{15}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{16}\! \left(x , y , z\right)\\ F_{16}\! \left(x , y , z\right) &= -\frac{-y F_{10}\! \left(x , y , z\right)+F_{10}\! \left(x , 1, z\right)}{-1+y}\\ \end{align*}