Av(1234, 1243, 1324, 1342, 2134, 2341, 3124)
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Generating Function
\(\displaystyle \frac{\left(x^{2}-x +1\right) \sqrt{1-4 x}+2 x^{3}-x^{2}+x -1}{2 \left(x -1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 17, 50, 154, 493, 1626, 5487, 18851, 65703, 231725, 825399, 2964951, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} F \left(x \right)^{2}-\left(x -1\right) \left(2 x^{3}-x^{2}+x -1\right) F \! \left(x \right)+x^{5}-x^{3}+2 x^{2}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(5+9 n \right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(12+5 n \right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 6\)
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This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 23 rules.

Found on January 21, 2022.

Finding the specification took 21 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)\\ F_{19}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{22}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{11}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Expand Verified" and has 63 rules.

Found on January 21, 2022.

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

This specification was found using the strategy pack "Insertion Row And Col Placements Req Corrob Expand Verified" and has 48 rules.

Found on January 21, 2022.

Finding the specification took 26 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{22}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{24} \left(x \right)^{2}\\ F_{38}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{47}\! \left(x \right) &= F_{7}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Expand Verified" and has 24 rules.

Found on January 21, 2022.

Finding the specification took 31 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{15}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= y x\\ F_{19}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{15}\! \left(x , y\right)\\ F_{20}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{23}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Expand Verified" and has 30 rules.

Found on January 21, 2022.

Finding the specification took 35 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_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \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_{9}\! \left(x \right)\\ \end{align*}\)