Av(1234, 1243, 1432, 2134, 2413)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right)}{x^{6}+x^{5}-x^{4}-x^{3}+x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 53, 146, 407, 1139, 3184, 8894, 24845, 69411, 193920, 541765, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}+x^{5}-x^{4}-x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)-a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{2247865}-\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{2247865}-\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +4}}{2247865}-\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +4}}{2247865}-\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +4}}{2247865}-\frac{63638 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +4}}{2247865}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{449573}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{449573}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{449573}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +3}}{449573}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +3}}{449573}-\frac{56031 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +3}}{449573}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{2247865}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{2247865}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{2247865}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{2247865}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +2}}{2247865}-\frac{148717 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +2}}{2247865}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{449573}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{449573}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{449573}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{449573}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{449573}+\frac{34574 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +1}}{449573}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n -1}}{2247865}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n -1}}{2247865}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n -1}}{2247865}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n -1}}{2247865}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n -1}}{2247865}+\frac{104947 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n -1}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{2247865}+\frac{368042 \mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{2247865}\)
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This specification was found using the strategy pack "Point Placements" and has 78 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_{19}\! \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_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \left(x \right)+F_{72}\! \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_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= x^{2}\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\ \end{align*}\)