Av(1243, 2413)
Counting Sequence
1, 1, 2, 6, 22, 88, 367, 1568, 6810, 29943, 132958, 595227, 2683373, 12170778, 55499358, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x F \left(x
\right)^{4}-\left(x -1\right)^{2} F \left(x
\right)^{3}+\left(3 x -2\right) \left(x -1\right) F \left(x
\right)^{2}+\left(x -1\right) F \! \left(x \right)+x = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 88\)
\(\displaystyle a \! \left(6\right) = 367\)
\(\displaystyle a \! \left(7\right) = 1568\)
\(\displaystyle a \! \left(n +8\right) = -\frac{500 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{100 \left(2 n +3\right) \left(35 n^{2}+129 n +124\right) a \! \left(n +1\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{5 \left(2017 n^{3}+15420 n^{2}+39683 n +34440\right) a \! \left(n +2\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{3 \left(2661 n^{3}+27572 n^{2}+95499 n +110908\right) a \! \left(n +3\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{3 \left(1297 n^{3}+17100 n^{2}+75071 n +109948\right) a \! \left(n +4\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{6 \left(n +6\right) \left(199 n^{2}+1988 n +4985\right) a \! \left(n +5\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(223 n^{2}+2615 n +7668\right) a \! \left(n +6\right)}{\left(n +8\right) \left(n +9\right)}+\frac{\left(23 n +147\right) a \! \left(n +7\right)}{n +9}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 88\)
\(\displaystyle a \! \left(6\right) = 367\)
\(\displaystyle a \! \left(7\right) = 1568\)
\(\displaystyle a \! \left(n +8\right) = -\frac{500 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{100 \left(2 n +3\right) \left(35 n^{2}+129 n +124\right) a \! \left(n +1\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{5 \left(2017 n^{3}+15420 n^{2}+39683 n +34440\right) a \! \left(n +2\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{3 \left(2661 n^{3}+27572 n^{2}+95499 n +110908\right) a \! \left(n +3\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{3 \left(1297 n^{3}+17100 n^{2}+75071 n +109948\right) a \! \left(n +4\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{6 \left(n +6\right) \left(199 n^{2}+1988 n +4985\right) a \! \left(n +5\right)}{\left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(223 n^{2}+2615 n +7668\right) a \! \left(n +6\right)}{\left(n +8\right) \left(n +9\right)}+\frac{\left(23 n +147\right) a \! \left(n +7\right)}{n +9}, \quad n \geq 8\)
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 Placements" and has 16 rules.
Found on April 25, 2021.Finding the specification took 3 seconds.
Copy 16 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{12}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 18 rules.
Found on April 25, 2021.Finding the specification took 4 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{14}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Point Placements" and has 26 rules.
Found on April 25, 2021.Finding the specification took 6 seconds.
Copy 26 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Point Placements Req Corrob" and has 25 rules.
Found on April 25, 2021.Finding the specification took 7 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)