Av(1234, 1432, 2143, 3241)
Generating Function
\(\displaystyle -\frac{4 x^{12}+10 x^{11}+8 x^{10}-2 x^{9}-17 x^{8}-11 x^{7}+10 x^{5}+6 x^{4}+x^{3}-2 x +1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 58, 145, 347, 818, 1911, 4400, 10027, 22698, 51110, 114557, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{12}+10 x^{11}+8 x^{10}-2 x^{9}-17 x^{8}-11 x^{7}+10 x^{5}+6 x^{4}+x^{3}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 145\)
\(\displaystyle a \! \left(7\right) = 347\)
\(\displaystyle a \! \left(8\right) = 818\)
\(\displaystyle a \! \left(9\right) = 1911\)
\(\displaystyle a \! \left(10\right) = 4400\)
\(\displaystyle a \! \left(11\right) = 10027\)
\(\displaystyle a \! \left(12\right) = 22698\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+8, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 145\)
\(\displaystyle a \! \left(7\right) = 347\)
\(\displaystyle a \! \left(8\right) = 818\)
\(\displaystyle a \! \left(9\right) = 1911\)
\(\displaystyle a \! \left(10\right) = 4400\)
\(\displaystyle a \! \left(11\right) = 10027\)
\(\displaystyle a \! \left(12\right) = 22698\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right)+8, \quad n \geq 13\)
Explicit Closed Form
\(\displaystyle \frac{5275423884 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +7}\right)}{1433463493}+\frac{1665548503 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +6}\right)}{130314863}+\frac{19629429390 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{1433463493}+\frac{4649179830 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{1433463493}-\frac{17897568714 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{1433463493}-\frac{19492604022 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{1433463493}-\frac{9469436798 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{1433463493}-\frac{10608399255 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{1433463493}+\frac{10412064148 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{9}+3 Z^{8}+2 Z^{7}-Z^{6}-4 Z^{5}-2 Z^{4}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{1433463493}-2 \left(\left\{\begin{array}{cc}-6 & n =0 \\ 3 & n =1 \\ -1 & n =2 \\ 2 & n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 123 rules.
Found on January 18, 2022.Finding the specification took 3 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_{28}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
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_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= x^{2}\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \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_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{90}\! \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_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{25}\! \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_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{63}\! \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_{86}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{67}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{77}\! \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_{70}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{18}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{77}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{18}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{18}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{118}\! \left(x \right)+F_{120}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{113}\! \left(x \right)+F_{88}\! \left(x \right)\\
\end{align*}\)