###### Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 24315, 24513, 25314, 25413, 31425, 31524, 32415, 32514)
Generating Function
$$\displaystyle -\frac{\left(2 x -1\right) \left(x +1\right) \left(x^{2}+x -1\right)}{6 x^{6}+4 x^{5}+6 x^{4}-4 x^{3}+x^{2}+3 x -1}$$
Counting Sequence
1, 1, 2, 6, 24, 80, 262, 820, 2582, 8130, 25728, 81434, 257854, 816112, 2582834, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(6 x^{6}+4 x^{5}+6 x^{4}-4 x^{3}+x^{2}+3 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x +1\right) \left(x^{2}+x -1\right) = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 24$$
$$\displaystyle a \! \left(5\right) = 80$$
$$\displaystyle a \! \left(n +6\right) = 6 a \! \left(n \right)+4 a \! \left(n +1\right)+6 a \! \left(n +2\right)-4 a \! \left(n +3\right)+a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6$$
Explicit Closed Form
$$\displaystyle -\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +4}}{4512811}-\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +4}}{4512811}-\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +4}}{4512811}-\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +4}}{4512811}-\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +4}}{4512811}-\frac{149020 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +4}}{4512811}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +3}}{13538433}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +3}}{13538433}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +3}}{13538433}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +3}}{13538433}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +3}}{13538433}-\frac{1644638 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +3}}{13538433}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +2}}{4512811}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +2}}{4512811}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +2}}{4512811}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +2}}{4512811}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +2}}{4512811}-\frac{613925 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +2}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +1}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +1}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +1}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +1}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +1}}{4512811}-\frac{19988 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +1}}{4512811}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n -1}}{13538433}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n -1}}{13538433}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n -1}}{13538433}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n -1}}{13538433}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n -1}}{13538433}+\frac{406109 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n -1}}{13538433}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n}}{4512811}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n}}{4512811}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n}}{4512811}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n}}{4512811}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n}}{4512811}+\frac{811180 \mathit{RootOf} \left(6 Z^{6}+4 Z^{5}+6 Z^{4}-4 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n}}{4512811}$$

### This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 86 rules.

Found on January 23, 2022.

Finding the specification took 12 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_{23}\! \left(x \right)+F_{6}\! \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_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= x^{2}\\ F_{38}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 0\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{59}\! \left(x \right)\\ \end{align*}