###### Av(12453, 12534, 12543, 13452, 13524, 13542, 14523, 14532, 21453, 21534, 21543, 23514, 24513, 31524, 32514)
Counting Sequence
1, 1, 2, 6, 24, 105, 478, 2237, 10707, 52202, 258410, 1295456, 6563870, 33561319, 172948169, ...
Implicit Equation for the Generating Function
$$\displaystyle x^{2} \left(x^{4}+x^{3}-x^{2}-4 x +4\right) F \left(x \right)^{3}+x \left(x^{3}+7 x^{2}-11 x +2\right) F \left(x \right)^{2}+\left(-3 x^{3}+6 x^{2}-1\right) F \! \left(x \right)-x^{2}-x +1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 24$$
$$\displaystyle a \! \left(5\right) = 105$$
$$\displaystyle a \! \left(6\right) = 478$$
$$\displaystyle a \! \left(7\right) = 2237$$
$$\displaystyle a \! \left(8\right) = 10707$$
$$\displaystyle a \! \left(9\right) = 52202$$
$$\displaystyle a \! \left(10\right) = 258410$$
$$\displaystyle a \! \left(11\right) = 1295456$$
$$\displaystyle a \! \left(12\right) = 6563870$$
$$\displaystyle a \! \left(13\right) = 33561319$$
$$\displaystyle a \! \left(14\right) = 172948169$$
$$\displaystyle a \! \left(15\right) = 897327948$$
$$\displaystyle a \! \left(16\right) = 4683652331$$
$$\displaystyle a \! \left(17\right) = 24576380580$$
$$\displaystyle a \! \left(18\right) = 129568975877$$
$$\displaystyle a \! \left(19\right) = 685997450424$$
$$\displaystyle a \! \left(20\right) = 3645876288227$$
$$\displaystyle a \! \left(21\right) = 19443934362723$$
$$\displaystyle a \! \left(n +22\right) = \frac{9 \left(3 n +5\right) \left(3 n +4\right) a \! \left(n \right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{\left(369 n^{2}+2457 n +3688\right) a \! \left(n +1\right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{3 \left(67 n^{2}+33 n -656\right) a \! \left(n +2\right)}{160 \left(n +23\right) \left(n +22\right)}-\frac{\left(2744 n^{2}+33471 n +98770\right) a \! \left(n +3\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(929 n^{2}+9578 n +35720\right) a \! \left(n +4\right)}{80 \left(n +23\right) \left(n +22\right)}+\frac{\left(1447 n^{2}+36129 n +102314\right) a \! \left(n +5\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(5903 n^{2}+15600 n -139652\right) a \! \left(n +6\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(13484 n^{2}+83163 n -217742\right) a \! \left(n +7\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(20909 n^{2}+244371 n +243100\right) a \! \left(n +8\right)}{480 \left(n +23\right) \left(n +22\right)}+\frac{\left(9667 n^{2}+67449 n +93164\right) a \! \left(n +9\right)}{480 \left(n +23\right) \left(n +22\right)}-\frac{\left(35247 n^{2}+552635 n +1943900\right) a \! \left(n +10\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(45192 n^{2}+804721 n +3237910\right) a \! \left(n +11\right)}{80 \left(n +23\right) \left(n +22\right)}-\frac{\left(228277 n^{2}+4673457 n +22605932\right) a \! \left(n +12\right)}{240 \left(n +23\right) \left(n +22\right)}+\frac{\left(245975 n^{2}+5646738 n +31324906\right) a \! \left(n +13\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(110825 n^{2}+2599392 n +14351614\right) a \! \left(n +14\right)}{240 \left(n +23\right) \left(n +22\right)}-\frac{\left(44510 n^{2}+1464093 n +12014848\right) a \! \left(n +15\right)}{120 \left(n +23\right) \left(n +22\right)}+\frac{\left(380171 n^{2}+12316059 n +99737140\right) a \! \left(n +16\right)}{480 \left(n +23\right) \left(n +22\right)}-\frac{\left(97265 n^{2}+3273035 n +27474928\right) a \! \left(n +17\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(31679 n^{2}+1079397 n +9104220\right) a \! \left(n +18\right)}{160 \left(n +23\right) \left(n +22\right)}+\frac{\left(3178 n^{2}+151065 n +1745060\right) a \! \left(n +19\right)}{120 \left(n +23\right) \left(n +22\right)}-\frac{\left(1677 n^{2}+70659 n +744136\right) a \! \left(n +20\right)}{40 \left(n +23\right) \left(n +22\right)}+\frac{\left(349 n +7466\right) a \! \left(n +21\right)}{30 n +690}, \quad n \geq 22$$

### This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 27 rules.

Found on January 22, 2022.

Finding the specification took 47 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= \frac{F_{12}\! \left(x , y\right) y -F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ \end{align*}