Av(1234, 1324, 2143, 2431, 4213)
Generating Function
\(\displaystyle \frac{3 x^{15}+7 x^{14}-4 x^{13}-21 x^{12}-11 x^{11}+24 x^{10}+27 x^{9}-20 x^{8}-22 x^{7}+10 x^{6}+9 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 112, 247, 539, 1143, 2371, 4833, 9713, 19295, 37960, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+3 x^{15}+7 x^{14}-4 x^{13}-21 x^{12}-11 x^{11}+24 x^{10}+27 x^{9}-20 x^{8}-22 x^{7}+10 x^{6}+9 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 112\)
\(\displaystyle a \! \left(7\right) = 247\)
\(\displaystyle a \! \left(8\right) = 539\)
\(\displaystyle a \! \left(9\right) = 1143\)
\(\displaystyle a \! \left(10\right) = 2371\)
\(\displaystyle a \! \left(11\right) = 4833\)
\(\displaystyle a \! \left(12\right) = 9713\)
\(\displaystyle a \! \left(13\right) = 19295\)
\(\displaystyle a \! \left(14\right) = 37960\)
\(\displaystyle a \! \left(15\right) = 74072\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+\frac{n}{4}-2, \quad n \geq 16\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 112\)
\(\displaystyle a \! \left(7\right) = 247\)
\(\displaystyle a \! \left(8\right) = 539\)
\(\displaystyle a \! \left(9\right) = 1143\)
\(\displaystyle a \! \left(10\right) = 2371\)
\(\displaystyle a \! \left(11\right) = 4833\)
\(\displaystyle a \! \left(12\right) = 9713\)
\(\displaystyle a \! \left(13\right) = 19295\)
\(\displaystyle a \! \left(14\right) = 37960\)
\(\displaystyle a \! \left(15\right) = 74072\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+\frac{n}{4}-2, \quad n \geq 16\)
Explicit Closed Form
\(\displaystyle \frac{97 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{44}+\frac{79 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{22}+\frac{195 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{44}+\frac{\left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{10}-\frac{\left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{10}-\frac{\left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{2}+\frac{623 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{50}+\frac{441 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{50}-\frac{889 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{50}+\left(\left\{\begin{array}{cc}-1 & n =0\text{ or } n =1\text{ or } n =2\text{ or } n =3 \\ 1 & n =4 \\ 4 & n =5 \\ 3 & n =6 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 85 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_{14}\! \left(x \right)+F_{7}\! \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_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \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_{6}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \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_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \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_{18}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{30}\! \left(x \right) &= 0\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \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_{7}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{53}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \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_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{77}\! \left(x \right)\\
\end{align*}\)