Av(132)
Generating Function
\(\displaystyle \frac{1-\sqrt{1-4 x}}{2 x}\)
Counting Sequence
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, ...
Implicit Equation for the Generating Function
\(\displaystyle x F \left(x
\right)^{2}-F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +2}, \quad n \geq 1\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +2}, \quad n \geq 1\)
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" and has 5 rules.
Found on May 17, 2021.Finding the specification took 0 seconds.
Copy 5 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
\end{align*}\)