Av(1234, 1243, 1324, 2314, 2341, 3124)
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Generating Function
\(\displaystyle \frac{-2 \sqrt{-4 x +1}\, x^{2}+4 x^{2}+\sqrt{-4 x +1}-1}{2 \left(x^{2}+x -1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 18, 56, 178, 579, 1923, 6506, 22365, 77933, 274718, 977979, 3511113, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} F \left(x \right)^{2}-\left(2 x +1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+4 x^{4}+3 x^{3}-4 x^{2}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(n +5\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{6 n a \! \left(1+n \right)}{n +6}-\frac{2 \left(13+7 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(8+n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 5\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 55 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= y x\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x , y\right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{33}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{21}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\ F_{48}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{50}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{48}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= \frac{F_{48}\! \left(x , y\right) y -F_{48}\! \left(x , 1\right)}{y -1}\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{y -1}\\ \end{align*}\)