Av(1243, 1324, 1342, 2314, 3124)
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Generating Function
\(\displaystyle \frac{\left(-2 x^{2}+2 x -1+\sqrt{-4 x +1}\right) \left(x -1\right)}{2 x^{2} \left(x^{2}-2 x +2\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 61, 200, 670, 2286, 7918, 27770, 98424, 351983, 1268541, 4602752, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{2}-2 x +2\right) F \left(x \right)^{2}+\left(x -1\right) \left(2 x^{2}-2 x +1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{\left(5+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(70+19 n \right) a \! \left(2+n \right)}{2 \left(6+n \right)}+\frac{\left(13 n +42\right) a \! \left(n +1\right)}{12+2 n}+\frac{2 \left(3 n +14\right) a \! \left(n +3\right)}{6+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 39 rules.

Found on July 23, 2021.

Finding the specification took 5 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_{24}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{24}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{31}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= y x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{24}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\ 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_{24}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{25}\! \left(x , y\right)+F_{7}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{10}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{22}\! \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_{10}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{22}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{20}\! \left(x , y\right) F_{22}\! \left(x \right) F_{24}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{24}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{0}\! \left(x \right) F_{24}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{24}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ \end{align*}\)