Av(1234, 1243, 2431, 3214)
Generating Function
\(\displaystyle -\frac{x^{11}+4 x^{10}+7 x^{9}+3 x^{8}-9 x^{7}-17 x^{6}-17 x^{5}-6 x^{4}+x^{2}+x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 59, 148, 358, 853, 2001, 4620, 10555, 23944, 53998, 121172, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{11}+4 x^{10}+7 x^{9}+3 x^{8}-9 x^{7}-17 x^{6}-17 x^{5}-6 x^{4}+x^{2}+x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 148\)
\(\displaystyle a \! \left(7\right) = 358\)
\(\displaystyle a \! \left(8\right) = 853\)
\(\displaystyle a \! \left(9\right) = 2001\)
\(\displaystyle a \! \left(10\right) = 4620\)
\(\displaystyle a \! \left(11\right) = 10555\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 148\)
\(\displaystyle a \! \left(7\right) = 358\)
\(\displaystyle a \! \left(8\right) = 853\)
\(\displaystyle a \! \left(9\right) = 2001\)
\(\displaystyle a \! \left(10\right) = 4620\)
\(\displaystyle a \! \left(11\right) = 10555\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 12\)
Explicit Closed Form
\(\displaystyle -\frac{114215103 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +6}\right)}{58508714}-\frac{256410353 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{29254357}-\frac{950602769 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{58508714}-\frac{1046188509 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{58508714}-\frac{316340961 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{29254357}-\frac{99767474 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{29254357}-\frac{1042851 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{58508714}+\frac{114588646 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{29254357}-\left(\left\{\begin{array}{cc}-6 & n =0 \\ 1 & n =1\text{ or } n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 68 rules.
Found on January 18, 2022.Finding the specification took 5 seconds.
Copy 68 equations to clipboard:
\(\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_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{54}\! \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_{33}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \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_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{40}\! \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_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{41}\! \left(x \right)+F_{50}\! \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_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{38}\! \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_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{58}\! \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_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{45}\! \left(x \right)+F_{65}\! \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_{12}\! \left(x \right)+F_{31}\! \left(x \right)\\
\end{align*}\)