Av(1243, 1324, 1342, 2314, 3124, 4123)
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Generating Function
\(\displaystyle -\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{4}+x^{3}+x^{2}-2 x +1\right)}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 18, 52, 155, 484, 1573, 5267, 18024, 62696, 220896, 786416, 2824170, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}-\left(x^{4}+x^{3}+x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x^{4}+x^{3}+x^{2}-2 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 155\)
\(\displaystyle a \! \left(7\right) = 484\)
\(\displaystyle a \! \left(8\right) = 1573\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(-12+n \right) a \! \left(1+n \right)}{7+n}-\frac{9 a \! \left(n +2\right)}{7+n}+\frac{3 \left(15+4 n \right) a \! \left(n +3\right)}{7+n}-\frac{3 \left(23+5 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(40+7 n \right) a \! \left(n +5\right)}{7+n}+\frac{4}{7+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 31 rules.

Found on January 21, 2022.

Finding the specification took 29 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{16}\! \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_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\ F_{11}\! \left(x , y\right) &= \frac{y F_{12}\! \left(x , y\right)-F_{12}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{3}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{28}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{29}\! \left(x \right) F_{3}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Expand Verified" and has 35 rules.

Found on January 21, 2022.

Finding the specification took 40 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_{17}\! \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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= -F_{6}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{24}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{30}\! \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_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \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_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob Expand Verified" and has 34 rules.

Found on January 21, 2022.

Finding the specification took 58 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_{17}\! \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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= -F_{6}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{29}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Expand Verified" and has 57 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_{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_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \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_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= -\frac{-y F_{26}\! \left(x , y\right)+F_{26}\! \left(x , 1\right)}{-1+y}\\ F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= y x\\ F_{29}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \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_{39}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{19}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{42}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{11}\! \left(x \right) F_{36}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{46}\! \left(x \right) &= -F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= -F_{54}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= \frac{F_{49}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= -F_{38}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{53}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{19}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{36} \left(x \right)^{2} F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Expand Verified" and has 34 rules.

Found on January 21, 2022.

Finding the specification took 59 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_{17}\! \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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= -F_{6}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{29}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)