Av(1234, 1342, 1423, 2431, 3214)
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Generating Function
\(\displaystyle \frac{x^{8}+3 x^{7}+6 x^{6}+11 x^{5}+5 x^{4}+x^{3}-x +1}{\left(x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 50, 111, 245, 546, 1209, 2655, 5829, 12812, 28151, 61822, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{8}+3 x^{7}+6 x^{6}+11 x^{5}+5 x^{4}+x^{3}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 111\)
\(\displaystyle a \! \left(7\right) = 245\)
\(\displaystyle a \! \left(8\right) = 546\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(4+n \right)+27, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+417 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+162\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+162 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+407\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{1-n}}{371}+\frac{\left(\left(\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-208\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{1-n}}{371}+\frac{\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}-363 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-242 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+299\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{1-n}}{371}+\frac{\left(\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2-n}}{371}+\frac{\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-363 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+43\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2-n}}{371}+\frac{\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3}}{371}+\frac{60 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{1-n}}{53}+\frac{285 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2-n}}{371}+\frac{309 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3}}{371}+\frac{121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{371}+\frac{\left(\left(\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-471 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-367\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-367 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-615\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(-54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-367 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-615\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-615 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-384\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{371}+\frac{\left(\left(\left(\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+208\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(\left(54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+205\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+208\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+208\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+208 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+240\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{371}+\frac{\left(\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+54\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+417 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+162\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}+417 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+404 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+108\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+54 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}+162 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+108 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+26\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{371}+\frac{\left(-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{4}-363 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}-242 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-121 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-28\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{371}+\left(\left\{\begin{array}{cc}5 & n =0 \\ 1 & n =1\text{ or } n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{93 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{371}-\frac{27}{7}\)
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This specification was found using the strategy pack "Point Placements" and has 72 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_{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_{4}\! \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_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{33}\! \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_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{40}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \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_{47}\! \left(x \right) &= F_{48}\! \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_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \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_{16}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{60}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \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_{14}\! \left(x \right)\\ \end{align*}\)