Av(1234, 1243, 2143, 2314, 3124)
Generating Function
\(\displaystyle \frac{x^{3}+x^{2}+2 x -1}{x^{3}+3 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+3 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}-\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(-2 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{16}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{16}\right)^{n}-\frac{2 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \sqrt{5}\, \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right)^{n}}{5}-\frac{\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}}{5}+\frac{\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\left(\mathrm{I} \sqrt{5}\, \sqrt{3}+\sqrt{5}+5\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \,\mathrm{I} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \sqrt{5}\, \sqrt{3}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}}{5}+\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \sqrt{3}}{8}-\frac{1}{8}\right)^{-n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\mathrm{I} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \sqrt{3}-\frac{\left(\mathrm{I} \sqrt{3}+\frac{3 \sqrt{5}}{5}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right)\right)}{24} & n <0 \\ 1 & n =0 \\ \frac{\left(4 \sqrt{5}\, \left(\left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-2 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(\left(8 \,\mathrm{I} \sqrt{3}-8\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-8 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{1024}-\frac{\mathrm{I} \sqrt{3}}{256}-\frac{1}{256}\right)^{-n} \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}+20 \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{128}-\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\left(8 \,\mathrm{I} \sqrt{3}-8\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}-8 \left(\mathrm{I} \sqrt{3}-2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}+1\right) \left(\sqrt{5}-1\right)\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}+2 \left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}}{240}\\-\\\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n} \left(\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}}{120} & 0<n \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 49 rules.
Found on January 18, 2022.Finding the specification took 0 seconds.
Copy 49 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_{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_{12}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \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_{30}\! \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_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{33}\! \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) &= 2 F_{13}\! \left(x \right)+F_{40}\! \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_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)\\
\end{align*}\)