###### Av(13254, 13524, 13542)
Counting Sequence
1, 1, 2, 6, 24, 117, 652, 3988, 26112, 180126, 1295090, 9631656, 73676572, 577180996, 4615090192, ...
Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point $$(i, j)$$ represents how many permutations have value $$j$$ at index $$i$$ (darker = more).

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Expand Verified" and has 30 rules.

Found on November 06, 2021.

Finding the specification took 816 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_{12}\! \left(x , y , z\right)+F_{5}\! \left(x , z\right)\\ F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right)\\ F_{16}\! \left(x , y , z\right) &= \frac{y z \left(F_{12}\! \left(x , y , z\right)-F_{12}\! \left(x , \frac{1}{z}, z\right)\right)}{y z -1}\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{9}\! \left(x , z\right)\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{22}\! \left(x , z , y\right)\\ F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x , y , z\right) &= \frac{y z F_{19}\! \left(x , y , z\right)-F_{19}\! \left(x , \frac{1}{z}, z\right)}{y z -1}\\ F_{22}\! \left(x , y , z\right) &= F_{19}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\ F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z , z\right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)\\ F_{25}\! \left(x , y , z\right) &= \frac{y F_{26}\! \left(x , y , 1\right)-z F_{26}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , y z \right)\\ F_{12}\! \left(x , y , z\right) &= F_{27}\! \left(x , y z , z\right)\\ F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y , z\right)\\ F_{29}\! \left(x , y , z\right) &= F_{12}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 38 rules.

Found on July 27, 2021.

Finding the specification took 5394 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_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{9}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{11}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{8}\! \left(x , y_{0}\right)\\ F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\ F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}, 1\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{21}\! \left(x , y_{0}, 1, y_{1}\right)-F_{21}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{1}\right)}{-1+y_{0}}\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{1}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{0}, y_{1}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{1}, y_{0}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\ F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}, y_{0}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{1}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, 1\right)-y_{1} F_{15}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\ F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{0}, y_{1}\right)\\ F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{15}\! \left(x , y_{0}, y_{1}\right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 32 rules.

Found on November 06, 2021.

Finding the specification took 3416 seconds.

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