Av(1234, 1243, 1432, 2134)
Generating Function
\(\displaystyle \frac{2 x^{3}+x^{2}+2 x -1}{2 x^{3}+3 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 204, 652, 2084, 6660, 21284, 68020, 217380, 694708, 2220164, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}+3 x -1\right) F \! \left(x \right)-2 x^{3}-x^{2}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -\frac{3 a \! \left(n +2\right)}{2}+\frac{a \! \left(n +3\right)}{2}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -\frac{3 a \! \left(n +2\right)}{2}+\frac{a \! \left(n +3\right)}{2}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-2-6 \,\mathrm{I}+\left(2+2 \,\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}-\frac{\left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}}}{8}\right)^{n} \sqrt{3}\, \left(\left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-2 \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}\right) \left(\frac{\left(1+3 \,\mathrm{I}-\left(1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}+2 \,\mathrm{I} \sqrt{3}+2}{-2+\left(\sqrt{3}-1\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n}}{36}\\+\\\frac{\left(\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2 \,\mathrm{I} \sqrt{3}-2\right)^{-n} \left(\left(\left(-1+\mathrm{I}+\left(-\frac{1}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\left(1-2 \,\mathrm{I}+\left(-\frac{2}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}\right) \left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}+\frac{\left(1+\sqrt{3}\right)^{\frac{2}{3}} 2^{\frac{2}{3}} \left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right)}{8}\right)^{n}-\left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\left(1+\mathrm{I}+\left(\frac{1}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\left(-1-2 \,\mathrm{I}+\left(\frac{2}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}-\frac{\left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}}}{8}\right)^{n}+2 \left(\frac{1}{-2+\left(\sqrt{3}-1\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n} \left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{32}+\frac{\mathrm{I} \sqrt{3}}{16}-\frac{1}{16}\right)^{-n} \left(\left(-2 \,\mathrm{I} \sqrt{3}+2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}+\left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}}\right)^{n} \left(\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}+\left(1+\sqrt{3}\right)^{\frac{2}{3}} 2^{\frac{2}{3}} \left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right)\right)^{n} 2^{-\frac{10 n}{3}} \left(1+\sqrt{3}\right)^{-\frac{n}{3}} \left(\left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}\right)\right)}{24} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 97 rules.
Found on January 18, 2022.Finding the specification took 5 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 97 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_{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_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{94}\! \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_{26}\! \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_{15}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \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_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= x^{2}\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \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_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{63}\! \left(x \right)+F_{71}\! \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_{68}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{76}\! \left(x \right)+F_{92}\! \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_{85}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \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_{4}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{76}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{72}\! \left(x \right)\\
\end{align*}\)