Av(132, 2134, 2314, 2341, 3124, 3241, 3412, 3421, 4123, 4213, 4231, 4321)
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Generating Function
\(\displaystyle \frac{2 x^{5}+2 x^{4}-3 x^{3}-x^{2}-1}{x -1}\)
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Counting Sequence
1, 1, 2, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(1-x \right) F \! \left(x \right)+2 x^{5}+2 x^{4}-3 x^{3}-x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 3\)
\(\displaystyle a \! \left(5\right) = 1\)
\(\displaystyle a \! \left(n \right) = 1, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}2 & n =2 \\ 5 & n =3 \\ 3 & n =4 \\ 1 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 25 rules.

Found on January 18, 2022.

Finding the specification took 0 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{4}\! \left(x \right)\\ \end{align*}\)