Av(1234, 1243, 2413, 3214, 3241)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(x^{8}+3 x^{7}-2 x^{6}-5 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1\right)}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 49, 120, 298, 732, 1788, 4350, 10556, 25572, 61878, 149616, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+\left(x +1\right) \left(x^{8}+3 x^{7}-2 x^{6}-5 x^{5}+4 x^{4}+x^{3}+2 x^{2}-3 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 298\)
\(\displaystyle a \! \left(8\right) = 732\)
\(\displaystyle a \! \left(9\right) = 1788\)
\(\displaystyle a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}, \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle \left(\left\{\begin{array}{cc}13 & n =0 \\ -5 & n =1 \\ 2 & n =2 \\ -2 & n =3 \\ 1 & n =4\text{ or } n =5 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{\left(-\sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\frac{\left(\sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\frac{\left(-20 \sqrt{2}-25\right) \left(-1-\sqrt{2}\right)^{-n}}{5}+\frac{\left(20 \sqrt{2}-25\right) \left(\sqrt{2}-1\right)^{-n}}{5}\)
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This specification was found using the strategy pack "Point Placements" and has 53 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{51}\! \left(x \right)\\ \end{align*}\)