Av(1243, 2314)
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Counting Sequence
1, 1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230, 13388094, 62688448, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{3} F \left(x \right)^{3}+\left(x^{2}-5 x +2\right) x F \left(x \right)^{2}+\left(x +1\right) \left(2 x -1\right) F \! \left(x \right)-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 88\)
\(\displaystyle a \! \left(6\right) = 367\)
\(\displaystyle a \! \left(7\right) = 1571\)
\(\displaystyle a \! \left(8\right) = 6861\)
\(\displaystyle a \! \left(9\right) = 30468\)
\(\displaystyle a \! \left(10\right) = 137229\)
\(\displaystyle a \! \left(11\right) = 625573\)
\(\displaystyle a \! \left(12\right) = 2881230\)
\(\displaystyle a \! \left(13\right) = 13388094\)
\(\displaystyle a \! \left(n +14\right) = \frac{18 n \left(n +1\right) a \! \left(n \right)}{\left(n +16\right) \left(n +15\right)}-\frac{18 \left(23 n +36\right) \left(n +1\right) a \! \left(n +1\right)}{\left(n +16\right) \left(n +15\right)}+\frac{9 \left(391 n^{2}+1620 n +1544\right) a \! \left(n +2\right)}{2 \left(n +16\right) \left(n +15\right)}-\frac{3 \left(433 n^{2}+286 n -4863\right) a \! \left(n +3\right)}{\left(n +16\right) \left(n +15\right)}-\frac{\left(18571 n^{2}+223276 n +638988\right) a \! \left(n +4\right)}{2 \left(n +16\right) \left(n +15\right)}+\frac{\left(266425 n^{2}+3417769 n +10815822\right) a \! \left(n +5\right)}{8 \left(n +16\right) \left(n +15\right)}-\frac{\left(455705 n^{2}+6632129 n +23950446\right) a \! \left(n +6\right)}{8 \left(n +16\right) \left(n +15\right)}+\frac{\left(245047 n^{2}+4033768 n +16509135\right) a \! \left(n +7\right)}{4 \left(n +16\right) \left(n +15\right)}-\frac{\left(358243 n^{2}+6605005 n +30306696\right) a \! \left(n +8\right)}{8 \left(n +16\right) \left(n +15\right)}+\frac{\left(91297 n^{2}+1867024 n +9507849\right) a \! \left(n +9\right)}{4 \left(n +16\right) \left(n +15\right)}-\frac{3 \left(21593 n^{2}+485627 n +2721148\right) a \! \left(n +10\right)}{8 \left(n +16\right) \left(n +15\right)}+\frac{\left(15593 n^{2}+382907 n +2343864\right) a \! \left(n +11\right)}{8 \left(n +16\right) \left(n +15\right)}-\frac{3 \left(401 n^{2}+10688 n +71049\right) a \! \left(n +12\right)}{4 \left(n +16\right) \left(n +15\right)}+\frac{\left(53 n +731\right) a \! \left(n +13\right)}{2 n +32}, \quad n \geq 14\)
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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 Tracked Fusion Isolated" and has 33 rules.

Found on April 23, 2021.

Finding the specification took 103 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_{26}\! \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_{32}\! \left(x \right)+F_{6}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{6}\! \left(x \right) &= 0\\ F_{7}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\ 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_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{30}\! \left(x , y\right)+F_{31}\! \left(x , y\right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{6}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{6}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{26}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= -\frac{y \left(F_{13}\! \left(x , 1\right)-F_{13}\! \left(x , y\right)\right)}{-1+y}\\ F_{26}\! \left(x \right) &= x\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= -\frac{y \left(F_{19}\! \left(x , 1\right)-F_{19}\! \left(x , y\right)\right)}{-1+y}\\ F_{30}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Isolated" and has 41 rules.

Found on April 21, 2021.

Finding the specification took 626 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_{21}\! \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_{21}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \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_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{21}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{21}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= x\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{21}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{21}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= 0\\ F_{35}\! \left(x \right) &= F_{21}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{21}\! \left(x \right) F_{22}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Isolated" and has 43 rules.

Found on April 23, 2021.

Finding the specification took 438 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_{23}\! \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_{23}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{40}\! \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_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{23}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{23}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{23}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{36}\! \left(x \right) &= 0\\ F_{37}\! \left(x \right) &= F_{23}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{23}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\ \end{align*}\)