Av(1234, 1243, 2134)
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Generating Function
\(\displaystyle \frac{-x \sqrt{-4 x +1}-4 x^{2}+9 x -2}{2 x^{3}-8 x^{2}+10 x -2}\)
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Counting Sequence
1, 1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-4 x^{2}+5 x -1\right) F \left(x \right)^{2}+\left(4 x -1\right) \left(x -2\right) F \! \left(x \right)+4 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{3+n}+\frac{3 \left(7+3 n \right) a \! \left(3+n \right)}{3+n}+\frac{\left(27+17 n \right) a \! \left(n +1\right)}{3+n}-\frac{6 \left(7+4 n \right) a \! \left(n +2\right)}{3+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion" and has 76 rules.

Found on July 23, 2021.

Finding the specification took 41 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_{19}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x , y\right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{22}\! \left(x \right) F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= y x\\ F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x , y\right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{23} \left(x \right)^{2}\\ F_{45}\! \left(x \right) &= F_{38}\! \left(x , 1\right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{57}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{7}\! \left(x \right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{71}\! \left(x \right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= \frac{F_{70}\! \left(x , y\right)-F_{70}\! \left(x , 1\right)}{-1+y}\\ F_{70}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\ F_{71}\! \left(x \right) &= F_{23}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{23} \left(x \right)^{2} F_{34}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= -\frac{y \left(F_{46}\! \left(x , 1\right)-F_{46}\! \left(x , y\right)\right)}{-1+y}\\ \end{align*}\)