Av(132, 4321)
Generating Function
$$\displaystyle -\frac{3 x^{4}-5 x^{3}+7 x^{2}-4 x +1}{\left(x -1\right)^{5}}$$
Counting Sequence
1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(x -1\right)^{5} F \! \left(x \right)+3 x^{4}-5 x^{3}+7 x^{2}-4 x +1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 5$$
$$\displaystyle a \! \left(4\right) = 13$$
$$\displaystyle a \! \left(n \right) = 1+\frac{11}{12} n^{2}-\frac{2}{3} n -\frac{1}{3} n^{3}+\frac{1}{12} n^{4}, \quad n \geq 5$$
Explicit Closed Form
$$\displaystyle 1+\frac{11}{12} n^{2}-\frac{2}{3} n -\frac{1}{3} n^{3}+\frac{1}{12} n^{4}$$

This specification was found using the strategy pack "Point Placements" and has 30 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{20}\! \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_{6}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}