1324 Domino
Counting Sequence
1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{4} F \left(x
\right)^{3}+2 x^{2} \left(3 x +1\right) F \left(x
\right)^{2}+\left(12 x^{2}-10 x +1\right) F \! \left(x \right)+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(n +1\right) = \frac{3 \left(3 n +5\right) \left(3 n +4\right) a \! \left(n \right)}{2 \left(n +3\right) \left(2 n +5\right)}, \quad n \geq 1\)
\(\displaystyle a \! \left(n +1\right) = \frac{3 \left(3 n +5\right) \left(3 n +4\right) a \! \left(n \right)}{2 \left(n +3\right) \left(2 n +5\right)}, \quad n \geq 1\)
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 33 rules.
Found on February 02, 2022.Finding the specification took 1872 seconds.
Copy 33 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{2}\! \left(x \right) &= 1\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1} \left(x \right)^{2} F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right) F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{2}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= y x\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right) F_{22}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{5}\! \left(x \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) &= F_{1}\! \left(x \right) F_{23}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{23}\! \left(x , y\right) F_{5}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right) F_{32}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\
\end{align*}\)