###### Av(12345, 12435, 12453, 13245, 13425, 13452, 14235, 14253, 14325, 14352, 14523, 14532, 31245, 31425, 31452)
Counting Sequence
1, 1, 2, 6, 24, 105, 474, 2194, 10392, 50185, 246194, 1223428, 6145544, 31154596, 159187944, ...
Implicit Equation for the Generating Function
$$\displaystyle x^{3} F \left(x \right)^{4}-4 x^{2} \left(x -1\right) F \left(x \right)^{3}+x \left(10 x^{2}-16 x +5\right) F \left(x \right)^{2}+\left(-6 x^{3}+9 x^{2}-2 x -2\right) F \! \left(x \right)+x^{3}-x^{2}-x +2 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 24$$
$$\displaystyle a \! \left(5\right) = 105$$
$$\displaystyle a \! \left(6\right) = 474$$
$$\displaystyle a \! \left(7\right) = 2194$$
$$\displaystyle a \! \left(8\right) = 10392$$
$$\displaystyle a \! \left(9\right) = 50185$$
$$\displaystyle a \! \left(10\right) = 246194$$
$$\displaystyle a \! \left(11\right) = 1223428$$
$$\displaystyle a \! \left(n +12\right) = \frac{11696 n \left(n +1\right) \left(n +2\right) a \! \left(n \right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}+\frac{24 \left(1189 n +10706\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}-\frac{24 \left(n +2\right) \left(31037 n^{2}+258453 n +496660\right) a \! \left(n +2\right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}+\frac{8 \left(433321 n^{3}+5434335 n^{2}+22021853 n +28861590\right) a \! \left(n +3\right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(694762 n^{3}+10422738 n^{2}+51455290 n +83539577\right) a \! \left(n +4\right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}+\frac{3 \left(4155343 n^{3}+72817569 n^{2}+423083962 n +814550396\right) a \! \left(n +5\right)}{2665 \left(n +13\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(25036681 n^{3}+502196757 n^{2}+3350434352 n +7432559910\right) a \! \left(n +6\right)}{5330 \left(n +13\right) \left(n +12\right) \left(n +11\right)}+\frac{3 \left(5843319 n^{3}+132090539 n^{2}+994210216 n +2491571848\right) a \! \left(n +7\right)}{5330 \left(n +13\right) \left(n +12\right) \left(n +11\right)}-\frac{3 \left(23063441 n^{3}+581215595 n^{2}+4877271630 n +13629925580\right) a \! \left(n +8\right)}{42640 \left(n +13\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(47282783 n^{3}+1318593687 n^{2}+12241724560 n +37840376508\right) a \! \left(n +9\right)}{85280 \left(n +13\right) \left(n +12\right) \left(n +11\right)}-\frac{3 \left(43316 n^{2}+851947 n +4188679\right) a \! \left(n +10\right)}{1040 \left(n +12\right) \left(n +13\right)}+\frac{3 \left(1461 n +15274\right) a \! \left(n +11\right)}{260 \left(n +13\right)}, \quad n \geq 12$$

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 23 rules.

Found on January 23, 2022.

Finding the specification took 184 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_{21}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{18}\! \left(x , y\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_{9}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right) F_{4}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)^{2} F_{12}\! \left(x , y\right) F_{4}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x \right) &= x\\ F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x \right)\\ \end{align*}

### This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 25 rules.

Found on January 22, 2022.

Finding the specification took 216 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_{23}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\ F_{5}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y , 1\right)\\ F_{7}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{8}\! \left(x , y , z\right)\\ F_{8}\! \left(x , y , z\right) &= F_{9}\! \left(x , y , z\right)\\ F_{9}\! \left(x , y , z\right) &= F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right) F_{4}\! \left(x , z\right)\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right)\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , y z \right)\\ F_{16}\! \left(x , y , z\right) &= F_{13}\! \left(x , z\right) F_{17}\! \left(x , y , z\right)\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right)+F_{6}\! \left(x , z\right)\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)\\ F_{19}\! \left(x , y , z\right) &= F_{10}\! \left(x , y\right) F_{10}\! \left(x , z\right) F_{13}\! \left(x , y\right) F_{4}\! \left(x , z\right)\\ F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right)\\ F_{21}\! \left(x , y , z\right) &= F_{10}\! \left(x , y\right) F_{22}\! \left(x , z\right) F_{23}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{23}\! \left(x \right)\\ \end{align*}