Av(1432, 2413)
Generating Function
\(\displaystyle \frac{-1+\sqrt{-8 x^{3}+16 x^{2}-8 x +1}}{4 x \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496, 66876096, ...
Implicit Equation for the Generating Function
\(\displaystyle 2 x \left(x -1\right) F \left(x
\right)^{2}+F \! \left(x \right)+x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{12 \left(3+2 n \right) a \! \left(1+n \right)}{5+n}-\frac{12 \left(5+2 n \right) a \! \left(n +2\right)}{5+n}+\frac{3 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{12 \left(3+2 n \right) a \! \left(1+n \right)}{5+n}-\frac{12 \left(5+2 n \right) a \! \left(n +2\right)}{5+n}+\frac{3 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
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 Row And Col Placements Req Corrob" and has 14 rules.
Found on April 26, 2021.Finding the specification took 41 seconds.
Copy 14 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{10}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements" and has 22 rules.
Found on April 26, 2021.Finding the specification took 14 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Row And Col Placements" and has 84 rules.
Found on April 26, 2021.Finding the specification took 29 seconds.
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Copy 84 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{0}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{2}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{2}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{2}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{8}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{2}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{72}\! \left(x \right)\\
\end{align*}\)