Av(1234, 1243, 1324, 1342, 1423, 2134)
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Generating Function
\(\displaystyle \frac{2 x^{3}+2 x^{2}-2 x +1-\sqrt{-4 x +1}}{2 x^{2} \left(x +2\right)}\)
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Counting Sequence
1, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, 91144, 325878, 1174281, 4260282, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x +2\right) F \left(x \right)^{2}+\left(-2 x^{3}-2 x^{2}+2 x -1\right) F \! \left(x \right)+x^{3}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +2\right) = \frac{\left(5+2 n \right) a \! \left(n \right)}{4+n}+\frac{\left(16+7 n \right) a \! \left(n +1\right)}{8+2 n}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 15 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{10}\! \left(x , y\right)-F_{10}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)