Av(1234, 1243, 1342, 2341, 3124, 4123)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{4} \sqrt{-4 x +1}+3 x^{4}-4 x^{3}+6 x^{2}-4 x +1}{2 x \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 18, 52, 152, 464, 1486, 4946, 16916, 58951, 208232, 743186, 2674804, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{8} F \left(x \right)^{2}-\left(3 x^{4}-4 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-6 x^{7}+24 x^{6}-50 x^{5}+66 x^{4}-55 x^{3}+28 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{n \left(n -1\right) \left(n^{2}-3 n -2\right)}{2 \left(n +2\right)}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 34 rules.

Found on January 20, 2022.

Finding the specification took 14 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_{19}\! \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_{19}\! \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_{19}\! \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_{16}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{19}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{33} \left(x \right)^{2} F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{19}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ \end{align*}\)