Av(1234, 1342, 2341, 3124, 4123)
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Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{5} \sqrt{-4 x +1}+x^{5}-7 x^{4}+10 x^{3}-10 x^{2}+5 x -1}{2 x \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 167, 499, 1556, 5072, 17126, 59281, 208727, 743901, 2675805, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{10} F \left(x \right)^{2}-\left(x^{5}-7 x^{4}+10 x^{3}-10 x^{2}+5 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{10}-10 x^{9}+44 x^{8}-114 x^{7}+200 x^{6}-242 x^{5}+205 x^{4}-119 x^{3}+45 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 167\)
\(\displaystyle a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{n \left(n -1\right) \left(3 n +2\right) \left(n -4\right) \left(n +1\right)}{24 \left(n +2\right)}, \quad n \geq 7\)

This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 71 rules.

Found on July 23, 2021.

Finding the specification took 10 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{2}\! \left(x \right) &= 1\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= x\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{51}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{53}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= y x\\ F_{60}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{57}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{62}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{3}\! \left(x \right)+F_{64}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{53}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= \frac{y F_{49}\! \left(x , y\right)-F_{49}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)