Av(12453, 12543, 13452, 13542, 23451, 23541)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3298, 18944, 111778, 673220, 4121434, 25570144, 160415810, 1015899124, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) F \left(x
\right)^{3}+\left(-3 x +1\right) F \left(x
\right)^{2}+2 F \! \left(x \right)-2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +5\right) = \frac{81 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +5\right) \left(n +4\right)}-\frac{9 \left(72 n^{2}+153 n +80\right) a \! \left(n +1\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{3 \left(n +2\right) \left(167 n +300\right) a \! \left(n +2\right)}{2 \left(n +5\right) \left(n +4\right)}-\frac{\left(185 n^{2}+1021 n +1400\right) a \! \left(n +3\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{\left(32 n +105\right) a \! \left(n +4\right)}{2 n +10}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +5\right) = \frac{81 n \left(2 n +1\right) a \! \left(n \right)}{\left(n +5\right) \left(n +4\right)}-\frac{9 \left(72 n^{2}+153 n +80\right) a \! \left(n +1\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{3 \left(n +2\right) \left(167 n +300\right) a \! \left(n +2\right)}{2 \left(n +5\right) \left(n +4\right)}-\frac{\left(185 n^{2}+1021 n +1400\right) a \! \left(n +3\right)}{2 \left(n +5\right) \left(n +4\right)}+\frac{\left(32 n +105\right) a \! \left(n +4\right)}{2 n +10}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 49 rules.
Found on January 23, 2022.Finding the specification took 176 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_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{24}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{27}\! \left(x , y\right) F_{28}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{25}\! \left(x , y\right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{24}\! \left(x \right) &= x\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= y x\\
F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{28}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{24}\! \left(x \right) F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= \frac{F_{43}\! \left(x , y\right) y -F_{43}\! \left(x , 1\right)}{-1+y}\\
F_{43}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{24}\! \left(x \right) F_{46}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 32 rules.
Found on January 22, 2022.Finding the specification took 11 seconds.
Copy 32 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y , 1\right)\\
F_{12}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{26}\! \left(x , y , z\right)\\
F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , y z \right)\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x , y , z\right) &= -\frac{-y F_{16}\! \left(x , y , z\right)+F_{16}\! \left(x , 1, z\right)}{-1+y}\\
F_{12}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , y z \right)\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{19}\! \left(x , y\right)\\
F_{18}\! \left(x , y , z\right) &= -\frac{-z F_{12}\! \left(x , y , z\right)+F_{12}\! \left(x , y , 1\right)}{-1+z}\\
F_{19}\! \left(x , y\right) &= y x\\
F_{20}\! \left(x , y , z\right) &= F_{19}\! \left(x , z\right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{25}\! \left(x , y\right) &= F_{16}\! \left(x , y , 1\right)\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , z\right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x , y , z\right) &= \frac{y z F_{12}\! \left(x , y , z\right)-F_{12}\! \left(x , y , \frac{1}{y}\right)}{y z -1}\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{29}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{21}\! \left(x , 1\right)\\
\end{align*}\)