Av(1234, 1243, 2314, 3412, 4123)
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Generating Function
\(\displaystyle -\frac{4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(-1+x \right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 144, 348, 793, 1733, 3678, 7650, 15695, 31907, 64476, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(-1+x \right)^{4} F \! \left(x \right)+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(2 n^{2}-n +6\right)}{6}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle 4 \,2^{n}-\frac{19 n}{6}-3-\frac{n^{2}}{2}-\frac{n^{3}}{3}\)
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This specification was found using the strategy pack "Row And Col Placements" and has 86 rules.

Found on July 23, 2021.

Finding the specification took 11 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{21}\! \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_{12}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{23} \left(x \right)^{2}\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{23}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{12}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{12}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{82}\! \left(x \right)\\ \end{align*}\)