Av(1234, 1342, 2341, 3412, 4123)
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Generating Function
\(\displaystyle -\frac{3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 151, 377, 886, 1989, 4316, 9136, 18997, 39006, 79389, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n +4\right) \left(n^{2}-n +6\right)}{24}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -4+5 \,2^{n}-\frac{15 n}{4}-\frac{23 n^{2}}{24}-\frac{n^{3}}{4}-\frac{n^{4}}{24}\)
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This specification was found using the strategy pack "Row And Col Placements" and has 49 rules.

Found on July 23, 2021.

Finding the specification took 9 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_{5}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{5}\! \left(x \right) &= x\\ 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_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{31}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ \end{align*}\)