Av(1243, 1342, 1423, 2134, 2314, 3124)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{5}-2 x^{3}+x^{2}-2 x +1}{2 \left(x^{7}+2 x^{6}+x^{5}+x^{4}-2 x +1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 18, 53, 163, 522, 1719, 5784, 19812, 68871, 242358, 861668, 3090432, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+2 x^{6}+x^{5}+x^{4}-2 x +1\right) x F \left(x \right)^{2}-\left(x -1\right) \left(2 x^{4}+2 x^{3}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 163\)
\(\displaystyle a \! \left(7\right) = 522\)
\(\displaystyle a \! \left(8\right) = 1719\)
\(\displaystyle a \! \left(n +9\right) = -\frac{2 \left(11+2 n \right) a \! \left(n \right)}{10+n}-\frac{3 \left(1+n \right) a \! \left(1+n \right)}{10+n}+5 a \! \left(n +2\right)-\frac{\left(1+n \right) a \! \left(n +3\right)}{10+n}+\frac{\left(31+4 n \right) a \! \left(n +4\right)}{10+n}-a \! \left(n +5\right)+\frac{4 \left(11+2 n \right) a \! \left(n +6\right)}{10+n}-\frac{2 \left(52+7 n \right) a \! \left(n +7\right)}{10+n}+\frac{\left(61+7 n \right) a \! \left(n +8\right)}{10+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 24 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{15}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= y x\\ F_{19}\! \left(x , y\right) &= \frac{y F_{20}\! \left(x , y\right)-F_{20}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= \frac{y F_{16}\! \left(x , y\right)-F_{16}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)