Av(1234, 1324, 1342, 4123, 4231)
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Generating Function
\(\displaystyle -\frac{\left(x^{2}-x +1\right) \left(2 x^{4}-7 x^{3}+10 x^{2}-5 x +1\right)}{\left(x -1\right)^{7}}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 135, 302, 617, 1171, 2092, 3554, 5787, 9088, 13833, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{7} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(2 x^{4}-7 x^{3}+10 x^{2}-5 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 135\)
\(\displaystyle a \! \left(n \right) = 1+\frac{167}{360} n^{2}-\frac{2}{5} n +\frac{5}{144} n^{4}-\frac{5}{48} n^{3}+\frac{1}{240} n^{5}+\frac{1}{720} n^{6}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle 1+\frac{167}{360} n^{2}-\frac{2}{5} n +\frac{5}{144} n^{4}-\frac{5}{48} n^{3}+\frac{1}{240} n^{5}+\frac{1}{720} n^{6}\)
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This specification was found using the strategy pack "Point Placements" and has 93 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{14}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{36}\! \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_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{46}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{66}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{46}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{84}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{89}\! \left(x \right)\\ \end{align*}\)