###### Av(12453, 12534, 12543, 21453, 21534, 21543, 24153, 24513, 24531, 25134, 25143, 25314, 25341, 25413, 25431)
Counting Sequence
1, 1, 2, 6, 24, 105, 480, 2254, 10776, 52182, 255120, 1256596, 6226176, 30998994, 154959938, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(3 x^{3}+4 x^{2}+20 x -4\right) F \left(x \right)^{3}-\left(x +3\right) \left(3 x^{3}+4 x^{2}+20 x -4\right) F \left(x \right)^{2}+\left(3 x^{5}+10 x^{4}+35 x^{3}+44 x^{2}+53 x -12\right) F \! \left(x \right)-3 x^{5}-7 x^{4}-19 x^{3}-17 x^{2}-17 x +4 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 24$$
$$\displaystyle a \! \left(5\right) = 105$$
$$\displaystyle a \! \left(6\right) = 480$$
$$\displaystyle a \! \left(n +6\right) = -\frac{3 \left(n -1\right) \left(n +1\right) a \! \left(n \right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{n \left(67 n +191\right) a \! \left(n +1\right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{5 \left(19 n +33\right) \left(n +1\right) a \! \left(n +2\right)}{2 \left(2 n +11\right) \left(n +5\right)}-\frac{\left(401 n^{2}+1569 n +1480\right) a \! \left(n +3\right)}{2 \left(2 n +11\right) \left(n +5\right)}+\frac{2 \left(37 n^{2}+209 n +285\right) a \! \left(n +4\right)}{\left(2 n +11\right) \left(n +5\right)}+\frac{\left(4 n^{2}+55 n +165\right) a \! \left(n +5\right)}{\left(2 n +11\right) \left(n +5\right)}, \quad n \geq 7$$

### This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 14 rules.

Found on January 23, 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x \right) &= x\\ \end{align*}