Av(1234, 1243, 1432, 3241, 4213)
Generating Function
\(\displaystyle \frac{x^{17}+5 x^{16}+19 x^{15}+37 x^{14}+35 x^{13}-11 x^{12}-66 x^{11}-87 x^{10}-53 x^{9}-8 x^{8}+31 x^{7}+31 x^{6}-6 x^{4}-x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 46, 79, 139, 279, 576, 1205, 2536, 5356, 11341, 24066, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{17}+5 x^{16}+19 x^{15}+37 x^{14}+35 x^{13}-11 x^{12}-66 x^{11}-87 x^{10}-53 x^{9}-8 x^{8}+31 x^{7}+31 x^{6}-6 x^{4}-x^{3}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 46\)
\(\displaystyle a \! \left(6\right) = 79\)
\(\displaystyle a \! \left(7\right) = 139\)
\(\displaystyle a \! \left(8\right) = 279\)
\(\displaystyle a \! \left(9\right) = 576\)
\(\displaystyle a \! \left(10\right) = 1205\)
\(\displaystyle a \! \left(11\right) = 2536\)
\(\displaystyle a \! \left(12\right) = 5356\)
\(\displaystyle a \! \left(13\right) = 11341\)
\(\displaystyle a \! \left(14\right) = 24066\)
\(\displaystyle a \! \left(15\right) = 51159\)
\(\displaystyle a \! \left(16\right) = 108910\)
\(\displaystyle a \! \left(17\right) = 232140\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-3 a \! \left(n +1\right)-5 a \! \left(n +2\right)-4 a \! \left(n +3\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+72, \quad n \geq 18\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 46\)
\(\displaystyle a \! \left(6\right) = 79\)
\(\displaystyle a \! \left(7\right) = 139\)
\(\displaystyle a \! \left(8\right) = 279\)
\(\displaystyle a \! \left(9\right) = 576\)
\(\displaystyle a \! \left(10\right) = 1205\)
\(\displaystyle a \! \left(11\right) = 2536\)
\(\displaystyle a \! \left(12\right) = 5356\)
\(\displaystyle a \! \left(13\right) = 11341\)
\(\displaystyle a \! \left(14\right) = 24066\)
\(\displaystyle a \! \left(15\right) = 51159\)
\(\displaystyle a \! \left(16\right) = 108910\)
\(\displaystyle a \! \left(17\right) = 232140\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-3 a \! \left(n +1\right)-5 a \! \left(n +2\right)-4 a \! \left(n +3\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+72, \quad n \geq 18\)
Explicit Closed Form
\(\displaystyle \frac{15269 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{7-n}\right)}{24552}+\frac{42767 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{6-n}\right)}{24552}+\frac{68755 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{5-n}\right)}{24552}+\frac{24203 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{4-n}\right)}{12276}+\frac{995 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{1364}-\frac{2839 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{2728}-\frac{4787 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{8184}-\frac{13549 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{12276}+\frac{21215 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+2 Z^{8}+2 Z^{7}-Z^{6}-2 Z^{5}-3 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{24552}+\left(\left\{\begin{array}{cc}-6 & n =0 \\ -7 & n =1\text{ or } n =2\text{ or } n =3 \\ -2 & n =4 \\ 10 & n =5 \\ 11 & n =6 \\ 3 & n =7 \\ 1 & n =8 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 103 rules.
Found on January 18, 2022.Finding the specification took 2 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_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \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_{29}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \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_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
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_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \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_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \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_{73}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \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_{69}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{92}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{15}\! \left(x \right)+F_{74}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)+F_{15}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 0\\
F_{101}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
\end{align*}\)