Av(13254, 13524, 13542, 31254, 31524, 31542, 35124, 35142, 35412)
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Counting Sequence
1, 1, 2, 6, 24, 111, 546, 2762, 14198, 73842, 387820, 2054640, 10971550, 59009651, 319465794, ...
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This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 29 rules.

Found on January 23, 2022.

Finding the specification took 48 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\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_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)^{2} F_{19}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 20 rules.

Found on January 22, 2022.

Finding the specification took 9 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= \frac{y F_{10}\! \left(x , 1, y\right)-F_{10}\! \left(x , \frac{1}{y}, y\right)}{-1+y}\\ F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\ F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right) F_{17}\! \left(x , z\right)\\ F_{12}\! \left(x , y , z\right) &= \frac{y F_{13}\! \left(x , y z , 1\right)-F_{13}\! \left(x , y z , \frac{1}{y}\right)}{-1+y}\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , y z \right)\\ F_{15}\! \left(x , y , z\right) &= F_{14}\! \left(x , y z , z\right)\\ F_{15}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)\\ F_{16}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{17}\! \left(x , z\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{19}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= -\frac{-y F_{10}\! \left(x , y , z\right)+F_{10}\! \left(x , 1, z\right)}{-1+y}\\ \end{align*}\)