Av(1234, 1243, 1324, 1342, 1423, 2134, 2314, 2341, 3124)
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Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{2}-x +1\right) \left(x +1\right)}{2 x}\)
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Counting Sequence
1, 1, 2, 6, 15, 44, 137, 443, 1472, 4994, 17225, 60216, 212874, 759696, 2733226, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{2}-\left(x +1\right) \left(x^{2}-x +1\right) F \! \left(x \right)+\left(x +1\right)^{2} \left(x^{2}-x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 15\)
\(\displaystyle a \! \left(5\right) = 44\)
\(\displaystyle a \! \left(6\right) = 137\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(-5+2 n \right) a \! \left(n \right)}{n -1}+\frac{2 \left(7+2 n \right) a \! \left(n +3\right)}{n -1}-\frac{\left(5+n \right) a \! \left(n +4\right)}{n -1}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 26 rules.

Found on January 20, 2022.

Finding the specification took 6 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= \frac{y F_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{25}\! \left(x \right) &= F_{18}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}\)