Av(1234, 2413)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(3 x -1\right)^{2}}{\left(x^{2}-4 x +1\right) \left(2 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 22, 86, 338, 1314, 5046, 19190, 72482, 272530, 1021734, 3823622, 14293234, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-4 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+\left(x -1\right) \left(3 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -4 a \! \left(n \right)+20 a \! \left(n +1\right)-21 a \! \left(n +2\right)+8 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -4 a \! \left(n \right)+20 a \! \left(n +1\right)-21 a \! \left(n +2\right)+8 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{\left(-2 \sqrt{3}+6\right) \left(2-\sqrt{3}\right)^{-n}}{18}+\frac{\left(2 \sqrt{3}+6\right) \left(2+\sqrt{3}\right)^{-n}}{18}-\frac{2^{n} \left(n -2\right)}{6}\)
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 "Regular Insertion Encoding Bottom" and has 55 rules.
Found on April 28, 2021.Finding the specification took 2 seconds.
Copy 55 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{13}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{50}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements" and has 61 rules.
Found on January 18, 2022.Finding the specification took 5 seconds.
Copy 61 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
\end{align*}\)