Av(12345, 12354, 12435, 13245, 13254, 21345, 21354, 21435)
Counting Sequence
1, 1, 2, 6, 24, 112, 567, 3012, 16516, 92614, 528045, 3049695, 17795682, 104723036, 620641217, ...
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 86 rules.
Finding the specification took 155 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_{20}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y_{0}\right)+F_{75}\! \left(x , y_{0}\right)+F_{83}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{8}\! \left(x , y_{0}\right)\\
F_{8}\! \left(x , y_{0}\right) &= F_{9}\! \left(x , y_{0}, 1\right)\\
F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}\right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{7}\! \left(x , y_{1}\right)+F_{73}\! \left(x , y_{0}, y_{1}\right)\\
F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{1}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{0}, y_{1}\right) F_{20}\! \left(x \right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{15}\! \left(x , y_{0}\right) y_{0}-F_{15}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{15}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y_{0}\right)+F_{17}\! \left(x , y_{0}\right)+F_{21}\! \left(x , y_{0}\right)+F_{23}\! \left(x , y_{0}\right)\\
F_{16}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{5}\! \left(x \right)\\
F_{17}\! \left(x , y_{0}\right) &= F_{18}\! \left(x , y_{0}\right) F_{20}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}\right) &= -\frac{-F_{19}\! \left(x , y_{0}\right) y_{0}+F_{19}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{19}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}, 1\right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{22}\! \left(x , y_{0}\right)\\
F_{22}\! \left(x , y_{0}\right) &= -\frac{-F_{15}\! \left(x , y_{0}\right) y_{0}+F_{15}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{23}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{24}\! \left(x , y_{0}\right)\\
F_{24}\! \left(x , y_{0}\right) &= F_{25}\! \left(x , y_{0}, 1\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-F_{26}\! \left(x , y_{0}, y_{1}\right) y_{1}+F_{26}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y_{0}, y_{1}\right)+F_{77}\! \left(x , y_{0}, y_{1}\right)+F_{78}\! \left(x , y_{0}, y_{1}\right)+F_{82}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}\right)\\
F_{28}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x , y_{0}\right)+F_{74}\! \left(x , y_{0}\right)+F_{75}\! \left(x , y_{0}\right)\\
F_{29}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{30}\! \left(x , y_{0}\right)\\
F_{30}\! \left(x , y_{0}\right) &= F_{31}\! \left(x , y_{0}, 1\right)\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{1}\right)+F_{33}\! \left(x , y_{0}, y_{1}\right)+F_{73}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{34}\! \left(x , y_{0}, y_{1}\right)\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y_{0}, y_{1}\right)+F_{37}\! \left(x , y_{1}\right)+F_{40}\! \left(x , y_{0}, y_{1}\right)+F_{42}\! \left(x , y_{0}, y_{1}\right)+F_{44}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{36}\! \left(x , y_{1}\right)\\
F_{36}\! \left(x , y_{0}\right) &= -\frac{-F_{6}\! \left(x , y_{0}\right) y_{0}+F_{6}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{37}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{38}\! \left(x , y_{0}\right)\\
F_{38}\! \left(x , y_{0}\right) &= F_{39}\! \left(x , y_{0}, 1\right)\\
F_{39}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{9}\! \left(x , y_{0}, y_{1}\right) y_{0} y_{1}-F_{9}\! \left(x , y_{0}, \frac{1}{y_{0}}\right)}{y_{0} y_{1}-1}\\
F_{40}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{41}\! \left(x , y_{0}, y_{1}\right)\\
F_{41}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{18}\! \left(x , y_{0}\right) y_{0}-F_{18}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{43}\! \left(x , y_{0}, y_{1}\right)\\
F_{43}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{22}\! \left(x , y_{0}\right) y_{0}-F_{22}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{45}\! \left(x , y_{0}, y_{1}\right)\\
F_{45}\! \left(x , y_{0}, y_{1}\right) &= F_{46}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{2}+F_{47}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\
F_{47}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{50}\! \left(x , y_{1}, y_{2}\right)+F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{72}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{49}\! \left(x , y_{1}, y_{2}\right)\\
F_{49}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{28}\! \left(x , y_{0}\right) y_{0}-F_{28}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{51}\! \left(x , y_{0}, y_{1}\right)\\
F_{51}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{31}\! \left(x , y_{0}, 1\right) y_{0}-F_{31}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{2}\right) F_{53}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{53}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{54}\! \left(x , y_{0}, y_{2}\right) y_{0}-F_{54}\! \left(x , y_{1}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{54}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{55}\! \left(x , 1, y_{1}\right) y_{1}-F_{55}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{55}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{56}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{20}\! \left(x \right) F_{57}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{57}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-F_{58}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{2}+F_{58}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\
F_{58}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{61}\! \left(x , y_{1}, y_{2}\right)+F_{63}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{67}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{72}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{59}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}\right) F_{60}\! \left(x , y_{1}, y_{2}\right)\\
F_{60}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{6}\! \left(x , y_{0}\right) y_{0}-F_{6}\! \left(x , y_{1}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{61}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{62}\! \left(x , y_{0}, y_{1}\right)\\
F_{62}\! \left(x , y_{0}, y_{1}\right) &= \frac{F_{9}\! \left(x , y_{0}, 1\right) y_{0}-F_{9}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{63}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{2}\right) F_{64}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{64}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{65}\! \left(x , y_{0}, y_{2}\right) y_{0}-F_{65}\! \left(x , y_{1}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{65}\! \left(x , y_{0}, y_{1}\right) &= -\frac{F_{66}\! \left(x , 1, y_{1}\right) y_{1}-F_{66}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right) y_{0}}{-y_{1}+y_{0}}\\
F_{66}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0} y_{1}, y_{1}\right)\\
F_{68}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{2}\right)+F_{11}\! \left(x , y_{1}, y_{2}\right)+F_{67}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{69}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{7}\! \left(x , y_{2}\right)\\
F_{68}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{10}\! \left(x , y_{0}, y_{2}\right) y_{0}-F_{10}\! \left(x , y_{1}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{69}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{20}\! \left(x \right) F_{70}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{70}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{71}\! \left(x , y_{0}, y_{2}\right) y_{0}-F_{71}\! \left(x , y_{1}, y_{2}\right) y_{1}}{-y_{1}+y_{0}}\\
F_{71}\! \left(x , y_{0}, y_{1}\right) &= F_{47}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{72}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{20}\! \left(x \right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{73}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{71}\! \left(x , y_{0}, y_{1}\right)\\
F_{74}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{36}\! \left(x , y_{0}\right)\\
F_{75}\! \left(x , y_{0}\right) &= F_{20}\! \left(x \right) F_{76}\! \left(x , y_{0}\right)\\
F_{76}\! \left(x , y_{0}\right) &= -\frac{-F_{28}\! \left(x , y_{0}\right) y_{0}+F_{28}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{77}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{54}\! \left(x , y_{0}, y_{1}\right)\\
F_{78}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{79}\! \left(x , y_{0}, y_{1}\right)\\
F_{79}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{80}\! \left(x , y_{0}, y_{1}\right)+F_{80}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\
F_{80}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y_{0}, y_{1}\right)+F_{13}\! \left(x , y_{0}, y_{1}\right)+F_{81}\! \left(x , y_{0}, y_{1}\right)+F_{82}\! \left(x , y_{0}, y_{1}\right)\\
F_{81}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{65}\! \left(x , y_{0}, y_{1}\right)\\
F_{82}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x \right) F_{25}\! \left(x , y_{0}, y_{1}\right)\\
F_{83}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}\right) F_{20}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{20}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\
\end{align*}\)