Av(1234)
Counting Sequence
1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, ...
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 "Row Placements Tracked Fusion" and has 18 rules.
Found on April 22, 2021.Finding the specification took 6 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{15}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , 1, y\right)\\
F_{7}\! \left(x , y , z\right) &= F_{8}\! \left(x , y z , z\right)\\
F_{8}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{13}\! \left(x , y , z\right)+F_{9}\! \left(x , y , z\right)\\
F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y , z\right)\\
F_{10}\! \left(x , y\right) &= y x\\
F_{11}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right) F_{12}\! \left(x , y , z\right)\\
F_{12}\! \left(x , y , z\right) &= \frac{-z F_{7}\! \left(x , 1, z\right)+y F_{7}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x , y , z\right) &= \frac{z F_{8}\! \left(x , y , z\right)-F_{8}\! \left(x , y , 1\right)}{-1+z}\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= \frac{y F_{4}\! \left(x , y\right)-F_{4}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion" and has 19 rules.
Found on April 23, 2021.Finding the specification took 16 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , 1, y\right)\\
F_{8}\! \left(x , y , z\right) &= F_{9}\! \left(x , y z , z\right)\\
F_{9}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y , z\right)+F_{12}\! \left(x , y , z\right)+F_{14}\! \left(x , y , z\right)\\
F_{10}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y , z\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y , z\right) &= F_{11}\! \left(x , z\right) F_{13}\! \left(x , y , z\right)\\
F_{13}\! \left(x , y , z\right) &= -\frac{z F_{8}\! \left(x , 1, z\right)-y F_{8}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{16}\! \left(x \right)\\
F_{15}\! \left(x , y , z\right) &= -\frac{-z F_{9}\! \left(x , y , z\right)+F_{9}\! \left(x , y , 1\right)}{-1+z}\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Col Placements Tracked Fusion" and has 63 rules.
Found on April 23, 2021.Finding the specification took 35 seconds.
Copy 63 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_{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_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{17}\! \left(x \right) &= 0\\
F_{18}\! \left(x , y\right) &= -\frac{y \left(F_{19}\! \left(x , 1\right)-F_{19}\! \left(x , y\right)\right)}{-1+y}\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , 1, y\right)\\
F_{24}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{11}\! \left(x , z\right) F_{24}\! \left(x , y , z\right)\\
F_{26}\! \left(x , y , z\right) &= F_{11}\! \left(x , z\right) F_{27}\! \left(x , y , z\right)\\
F_{27}\! \left(x , y , z\right) &= F_{28}\! \left(x , y z , z\right)\\
F_{28}\! \left(x , y , z\right) &= -\frac{z F_{24}\! \left(x , 1, z\right)-y F_{24}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{29}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{33}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , 1, y\right)\\
F_{36}\! \left(x , y , z\right) &= F_{37}\! \left(x , z\right)+F_{40}\! \left(x , y , z\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x , y\right)\\
F_{40}\! \left(x , y , z\right) &= F_{17}\! \left(x \right)+F_{41}\! \left(x , y , z\right)+F_{44}\! \left(x , z , y\right)\\
F_{41}\! \left(x , y , z\right) &= \frac{z \left(F_{42}\! \left(x , y , 1\right)-F_{42}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\
F_{42}\! \left(x , y , z\right) &= F_{43}\! \left(x , y , y z \right)\\
F_{43}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{36}\! \left(x , y , z\right)\\
F_{44}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{45}\! \left(x , z , y\right)\\
F_{45}\! \left(x , y , z\right) &= F_{21}\! \left(x , y\right)+F_{40}\! \left(x , y , z\right)\\
F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , 1, y\right)\\
F_{47}\! \left(x , y , z\right) &= F_{32}\! \left(x , z\right)+F_{48}\! \left(x , y , z\right)\\
F_{48}\! \left(x , y , z\right) &= F_{17}\! \left(x \right)+F_{49}\! \left(x , y , z\right)+F_{55}\! \left(x , y , z\right)+F_{58}\! \left(x , z , y\right)\\
F_{49}\! \left(x , y , z\right) &= -\frac{y \left(F_{50}\! \left(x , 1, z\right)-F_{50}\! \left(x , y , z\right)\right)}{-1+y}\\
F_{51}\! \left(x , y , z\right) &= F_{50}\! \left(x , y z , z\right)\\
F_{51}\! \left(x , y , z\right) &= F_{4}\! \left(x \right) F_{52}\! \left(x , y , z\right)\\
F_{52}\! \left(x , y , z\right) &= F_{53}\! \left(x , y , z\right)+F_{54}\! \left(x , y , z\right)\\
F_{53}\! \left(x , y , z\right) &= F_{40}\! \left(x , y z , z\right)\\
F_{54}\! \left(x , y , z\right) &= F_{48}\! \left(x , y z , z\right)\\
F_{55}\! \left(x , y , z\right) &= \frac{z \left(F_{56}\! \left(x , y , 1\right)-F_{56}\! \left(x , y , \frac{z}{y}\right)\right)}{-z +y}\\
F_{56}\! \left(x , y , z\right) &= F_{57}\! \left(x , y , y z \right)\\
F_{57}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{47}\! \left(x , y , z\right)\\
F_{58}\! \left(x , y , z\right) &= F_{11}\! \left(x , y\right) F_{59}\! \left(x , z , y\right)\\
F_{59}\! \left(x , y , z\right) &= F_{16}\! \left(x , y\right)+F_{48}\! \left(x , y , z\right)\\
F_{60}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , 1, y\right)\\
F_{62}\! \left(x , y , z\right) &= F_{59}\! \left(x , y z , z\right)\\
\end{align*}\)