Av(1234, 1243, 2431, 3214, 4132)
Generating Function
\(\displaystyle \frac{4 x^{15}+27 x^{14}+79 x^{13}+131 x^{12}+127 x^{11}+58 x^{10}-20 x^{9}-55 x^{8}-49 x^{7}-21 x^{6}+7 x^{5}+5 x^{4}-x^{2}-x +1}{\left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 47, 85, 161, 333, 705, 1502, 3219, 6921, 14919, 32251, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+4 x^{15}+27 x^{14}+79 x^{13}+131 x^{12}+127 x^{11}+58 x^{10}-20 x^{9}-55 x^{8}-49 x^{7}-21 x^{6}+7 x^{5}+5 x^{4}-x^{2}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(6\right) = 85\)
\(\displaystyle a \! \left(7\right) = 161\)
\(\displaystyle a \! \left(8\right) = 333\)
\(\displaystyle a \! \left(9\right) = 705\)
\(\displaystyle a \! \left(10\right) = 1502\)
\(\displaystyle a \! \left(11\right) = 3219\)
\(\displaystyle a \! \left(12\right) = 6921\)
\(\displaystyle a \! \left(13\right) = 14919\)
\(\displaystyle a \! \left(14\right) = 32251\)
\(\displaystyle a \! \left(15\right) = 69894\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 16\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(6\right) = 85\)
\(\displaystyle a \! \left(7\right) = 161\)
\(\displaystyle a \! \left(8\right) = 333\)
\(\displaystyle a \! \left(9\right) = 705\)
\(\displaystyle a \! \left(10\right) = 1502\)
\(\displaystyle a \! \left(11\right) = 3219\)
\(\displaystyle a \! \left(12\right) = 6921\)
\(\displaystyle a \! \left(13\right) = 14919\)
\(\displaystyle a \! \left(14\right) = 32251\)
\(\displaystyle a \! \left(15\right) = 69894\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 16\)
Explicit Closed Form
\(\displaystyle \frac{56305 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{6-n}\right)}{3079406}+\frac{400253 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{5-n}\right)}{3079406}+\frac{701601 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{4-n}\right)}{3079406}+\frac{233626 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{1539703}+\frac{606367 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{3079406}+\frac{243796 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{1539703}+\frac{612499 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{3079406}+\frac{209105 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{8}+4 Z^{7}+6 Z^{6}+5 Z^{5}+Z^{4}-Z^{3}-Z^{2}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{3079406}+\left(\left\{\begin{array}{cc}-1 & n =1 \\ -2 & n =2\text{ or } n =3 \\ 1 & n =4 \\ 11 & n =5\text{ or } n =6 \\ 4 & n =7 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 96 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_{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_{20}\! \left(x \right)+F_{51}\! \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_{26}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{39}\! \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_{20}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{38}\! \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_{49}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{52}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{56}\! \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_{61}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \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_{36}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{40}\! \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_{73}\! \left(x \right)+F_{76}\! \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_{30}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \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_{87}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{38}\! \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_{4}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{40}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{89}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{90}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
\end{align*}\)