Av(1234, 1324, 1342, 1423, 2314, 3124)
Generating Function
\(\displaystyle -\frac{\left(\frac{\left(x -1\right)^{2} \sqrt{-4 x +1}}{2}+\left(x^{2}+1\right) \left(x -\frac{1}{2}\right)\right) \left(x -1\right)^{2}}{x \left(x^{5}-2 x^{3}+5 x^{2}-4 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 18, 54, 167, 534, 1755, 5896, 20167, 70014, 246105, 874175, 3132871, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{5}-2 x^{3}+5 x^{2}-4 x +1\right) F \left(x
\right)^{2}+\left(2 x -1\right) \left(x^{2}+1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 167\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{8+n}+\frac{\left(5 n +29\right) a \! \left(1+n \right)}{8+n}+\frac{\left(7 n +4\right) a \! \left(n +2\right)}{8+n}-\frac{2 \left(15 n +44\right) a \! \left(n +3\right)}{8+n}+\frac{\left(43 n +185\right) a \! \left(n +4\right)}{8+n}-\frac{\left(162+29 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(9 n +61\right) a \! \left(n +6\right)}{8+n}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 167\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(2 n +3\right) a \! \left(n \right)}{8+n}+\frac{\left(5 n +29\right) a \! \left(1+n \right)}{8+n}+\frac{\left(7 n +4\right) a \! \left(n +2\right)}{8+n}-\frac{2 \left(15 n +44\right) a \! \left(n +3\right)}{8+n}+\frac{\left(43 n +185\right) a \! \left(n +4\right)}{8+n}-\frac{\left(162+29 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(9 n +61\right) a \! \left(n +6\right)}{8+n}, \quad n \geq 7\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 19 rules.
Found on July 23, 2021.Finding the specification took 13 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= y x\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17} \left(x \right)^{2} F_{0}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
\end{align*}\)