Av(1243, 1324, 3214, 3421)
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Generating Function
\(\displaystyle \frac{x^{7}-4 x^{6}+9 x^{4}-11 x^{3}+11 x^{2}-5 x +1}{\left(x -1\right)^{6}}\)
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Counting Sequence
1, 1, 2, 6, 20, 61, 157, 349, 693, 1262, 2148, 3464, 5346, 7955, 11479, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right)^{6} F \! \left(x \right)+x^{7}-4 x^{6}+9 x^{4}-11 x^{3}+11 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 157\)
\(\displaystyle a \! \left(7\right) = 349\)
\(\displaystyle a \! \left(n \right) = \frac{\left(-1+n \right) \left(2 n^{4}+17 n^{3}-103 n^{2}+182 n +120\right)}{120}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -1-n^{3}+\frac{19}{8} n^{2}-\frac{31}{60} n +\frac{1}{60} n^{5}+\frac{1}{8} n^{4} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 85 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_{18}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \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_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{35}\! \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_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \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_{48}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{49}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{40}\! \left(x \right)\\ \end{align*}\)