Av(1234, 1342, 1423, 2341)
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Generating Function
\(\displaystyle \frac{-4 \left(x -\frac{1}{2}\right)^{2} \sqrt{-4 x +1}-2 x^{3}+10 x^{2}-6 x +1}{2 x^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{3} F \left(x \right)^{2}+\left(2 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +2\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{6 \left(3+n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 58 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \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_{8}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= y x\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{12}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{13}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)+F_{51}\! \left(x \right)\\ F_{19}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x , y\right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\ F_{34}\! \left(x \right) &= 0\\ F_{35}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x , y\right)\\ F_{39}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{44}\! \left(x , y\right) &= 2 F_{34}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{50}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{55}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\ \end{align*}\)