Av(1243, 1324, 1432, 2341, 3241)
Generating Function
\(\displaystyle -\frac{x^{8}+4 x^{7}-x^{6}-7 x^{5}-x^{4}-2 x^{3}+8 x^{2}-5 x +1}{\left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 135, 330, 790, 1875, 4442, 10536, 25051, 59723, 142739, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}+4 x^{7}-x^{6}-7 x^{5}-x^{4}-2 x^{3}+8 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 135\)
\(\displaystyle a \! \left(7\right) = 330\)
\(\displaystyle a \! \left(8\right) = 790\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{2}}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}-\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}-\frac{9 n}{2}+\frac{5}{2}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 135\)
\(\displaystyle a \! \left(7\right) = 330\)
\(\displaystyle a \! \left(8\right) = 790\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{2}}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}-\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}-\frac{9 n}{2}+\frac{5}{2}, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-36 \sqrt{11}+176 \,\mathrm{I}\right) \sqrt{3}-108 \,\mathrm{I} \sqrt{11}+176\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+352+\left(\left(143 \,\mathrm{I}+21 \sqrt{11}\right) \sqrt{3}-63 \,\mathrm{I} \sqrt{11}-143\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1056}\\+\\\frac{\left(\left(\left(-143 \,\mathrm{I}+21 \sqrt{11}\right) \sqrt{3}+63 \,\mathrm{I} \sqrt{11}-143\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+352+\left(\left(-176 \,\mathrm{I}-36 \sqrt{11}\right) \sqrt{3}+108 \,\mathrm{I} \sqrt{11}+176\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1056}\\+\\\frac{\left(\left(-42 \sqrt{11}\, \sqrt{3}+286\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+72 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-352 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+352\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{1056}\\+\frac{\left(-264 \sqrt{2}+264\right) \left(-1-\sqrt{2}\right)^{-n}}{1056}+\frac{\left(264 \sqrt{2}+264\right) \left(\sqrt{2}-1\right)^{-n}}{1056}-\frac{n^{2}}{4}\\-\frac{9 n}{4}+\frac{1}{2} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 92 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \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_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \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_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= x^{2}\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{52}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{55}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{62}\! \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_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= x^{2}\\
F_{82}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{87}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{75}\! \left(x \right)\\
\end{align*}\)