Av(1234, 1243, 1324, 1342, 3124)
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Generating Function
\(\displaystyle \frac{\left(-x^{2}+x -1\right) \sqrt{1-4 x}-2 x^{4}+6 x^{3}-9 x^{2}+7 x -1}{2 x^{5}-8 x^{4}+16 x^{3}-18 x^{2}+12 x -2}\)
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Counting Sequence
1, 1, 2, 6, 19, 62, 209, 722, 2538, 9040, 32539, 118135, 431997, 1589435, 5878927, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-4 x^{4}+8 x^{3}-9 x^{2}+6 x -1\right) F \left(x \right)^{2}+\left(2 x^{4}-6 x^{3}+9 x^{2}-7 x +1\right) F \! \left(x \right)+x \left(x^{2}-2 x +2\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 209\)
\(\displaystyle a \! \left(7\right) = 722\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{n +8}+\frac{\left(62+21 n \right) a \! \left(n +1\right)}{n +8}-\frac{\left(200+57 n \right) a \! \left(n +2\right)}{n +8}+\frac{\left(402+97 n \right) a \! \left(n +3\right)}{n +8}-\frac{\left(546+113 n \right) a \! \left(n +4\right)}{n +8}+\frac{\left(494+87 n \right) a \! \left(n +5\right)}{n +8}-\frac{22 \left(13+2 n \right) a \! \left(n +6\right)}{n +8}+\frac{\left(80+11 n \right) a \! \left(n +7\right)}{n +8}, \quad n \geq 8\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 26 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \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_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{13}\! \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_{9}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \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_{10}\! \left(x \right) F_{20}\! \left(x , y\right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ \end{align*}\)