Av(123, 132, 213, 321)
Generating Function
\(\displaystyle x^{4}+2 x^{3}+2 x^{2}+x +1\)
Counting Sequence
1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
Implicit Equation for the Generating Function
\(\displaystyle -F \! \left(x \right)+x^{4}+2 x^{3}+2 x^{2}+x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 2\)
\(\displaystyle a \! \left(4\right) = 1\)
\(\displaystyle a \! \left(n \right) = 0, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 2\)
\(\displaystyle a \! \left(4\right) = 1\)
\(\displaystyle a \! \left(n \right) = 0, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2\text{ or } n =3 \\ 1 & n =4 \\ 0 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 12 rules.
Found on January 18, 2022.Finding the specification took 0 seconds.
Copy 12 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)