###### Av(123456)
Counting Sequence
1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, ...

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 55 rules.

Found on January 29, 2022.

Finding the specification took 312 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_{38}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y_{0}\right)+F_{53}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}, 1\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{8}\! \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_{51}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{9}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{12}\! \left(x , y_{0}, y_{1} y_{2}, y_{2} y_{3}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{2}, y_{1}\right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{24}\! \left(x , y_{1}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{17}\! \left(x , y_{0}, y_{1}, y_{2}, y_{2} y_{3}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{19}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{20}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}, y_{3}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{27}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{33}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)+F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{22}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}, y_{3}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{0}\right) F_{25}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{24}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{26}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}, y_{3}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{1}\right) F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-F_{19}\! \left(x , 1, y_{1}, y_{2}, y_{3}\right) y_{1}+F_{19}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}, y_{3}\right) y_{0}}{-y_{1}+y_{0}}\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{3}\right) F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{-F_{31}\! \left(x , y_{0}, 1, y_{2}, y_{3}\right) y_{2} y_{3}+F_{31}\! \left(x , y_{0}, \frac{y_{1}}{y_{2} y_{3}}, y_{2}, y_{3}\right) y_{1}}{-y_{2} y_{3}+y_{1}}\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{32}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{20}\! \left(x , y_{0}, y_{1} y_{2}, y_{3}, y_{2}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{3}\right) F_{34}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) y_{2}-F_{20}\! \left(x , y_{0}, y_{1}, 1, y_{3}\right)}{-1+y_{2}}\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) F_{38}\! \left(x \right)\\ F_{36}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, 1\right) y_{3}-F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, \frac{1}{y_{3}}\right)}{-1+y_{3}}\\ F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, y_{4}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}, y_{3} y_{4}\right)\\ F_{38}\! \left(x \right) &= x\\ F_{39}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{41}\! \left(x , \frac{y_{0}}{y_{2}}, y_{2}, y_{1}\right) y_{0} y_{1}-F_{41}\! \left(x , \frac{1}{y_{1}}, y_{2}, y_{1}\right) y_{2}}{y_{0} y_{1}-y_{2}}\\ F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0} y_{1}, y_{2}, y_{1}\right)\\ F_{42}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{1}\right) F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{43}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{44}\! \left(x , y_{0}, y_{1}, 1\right) y_{1}-F_{44}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{1}}\right) y_{2}}{y_{1}-y_{2}}\\ F_{44}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, y_{1} y_{2}\right)\\ F_{45}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x \right) F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{46}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, 1\right) y_{1}-F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, \frac{1}{y_{1}}\right)}{-1+y_{1}}\\ F_{47}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{2}, y_{1} y_{3}\right)\\ F_{48}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{2}\right) F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{49}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-F_{50}\! \left(x , 1, y_{1} y_{2}\right) y_{2}+F_{50}\! \left(x , \frac{y_{0}}{y_{2}}, y_{1} y_{2}\right) y_{0}}{-y_{2}+y_{0}}\\ F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0}, y_{1}, 1\right)\\ F_{51}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x \right) F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{52}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{F_{8}\! \left(x , y_{0}, y_{1}, y_{2}\right) y_{1} y_{2}-F_{8}\! \left(x , y_{0}, y_{1}, \frac{1}{y_{1}}\right)}{y_{1} y_{2}-1}\\ F_{53}\! \left(x , y_{0}\right) &= F_{38}\! \left(x \right) F_{54}\! \left(x , y_{0}\right)\\ F_{54}\! \left(x , y_{0}\right) &= \frac{F_{4}\! \left(x , y_{0}\right) y_{0}-F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}