Av(1243, 2134)
Generating Function
\(\displaystyle \frac{\left(-x^{2}+x \right) \sqrt{-4 x +1}+3 x^{2}-9 x +2}{2 \left(x^{2}+4 x -1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 22, 87, 354, 1459, 6056, 25252, 105632, 442916, 1860498, 7826120, 32956964, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+4 x -1\right) \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(3 x^{2}-9 x +2\right) F \! \left(x \right)+x^{3}-4 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(n +3\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{2+n}+\frac{2 \left(5+4 n \right) a \! \left(2+n \right)}{2+n}-\frac{3 \left(2+5 n \right) a \! \left(n +1\right)}{2+n}+\frac{6 n}{2+n}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(n +3\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{2+n}+\frac{2 \left(5+4 n \right) a \! \left(2+n \right)}{2+n}-\frac{3 \left(2+5 n \right) a \! \left(n +1\right)}{2+n}+\frac{6 n}{2+n}, \quad n \geq 5\)
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Isolated" and has 30 rules.
Found on April 22, 2021.Finding the specification took 779 seconds.
Copy 30 equations to clipboard:
\(\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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \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_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x , y\right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= y x\\
F_{28}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= -\frac{y \left(F_{24}\! \left(x , 1\right)-F_{24}\! \left(x , y\right)\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Partial Col Placements Tracked Fusion Isolated" and has 94 rules.
Found on April 22, 2021.Finding the specification took 1673 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_{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_{28}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x , y\right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x \right) F_{49}\! \left(x , y\right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= y x\\
F_{52}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{49}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x , y\right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{46} \left(x \right)^{2}\\
F_{59}\! \left(x \right) &= F_{53}\! \left(x , 1\right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{67}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{70}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{78}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{51}\! \left(x , y\right) F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)+F_{89}\! \left(x \right)\\
F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{82}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= -\frac{F_{88}\! \left(x , 1\right)-F_{88}\! \left(x , y\right)}{-1+y}\\
F_{88}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{89}\! \left(x \right) &= F_{32}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{46} \left(x \right)^{2} F_{49}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= -\frac{y \left(F_{60}\! \left(x , 1\right)-F_{60}\! \left(x , y\right)\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Isolated" and has 33 rules.
Found on April 25, 2021.Finding the specification took 798 seconds.
Copy 33 equations to clipboard:
\(\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_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \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_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{7}\! \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_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= y x\\
F_{30}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= -\frac{y \left(F_{26}\! \left(x , 1\right)-F_{26}\! \left(x , y\right)\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion Isolated" and has 98 rules.
Found on April 20, 2021.Finding the specification took 793 seconds.
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Copy 98 equations to clipboard:
\(\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_{27}\! \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_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \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_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \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_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{49}\! \left(x \right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x \right) F_{45}\! \left(x , y\right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{45}\! \left(x , y\right) F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= y x\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{42} \left(x \right)^{2}\\
F_{61}\! \left(x \right) &= F_{54}\! \left(x , 1\right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{64}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{69}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{72}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{80}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{48}\! \left(x , y\right) F_{86}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)+F_{93}\! \left(x \right)\\
F_{87}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{84}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{42} \left(x \right)^{2} F_{45}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= -\frac{F_{62}\! \left(x , 1\right)-F_{62}\! \left(x , y\right)}{-1+y}\\
F_{93}\! \left(x \right) &= F_{31}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{42} \left(x \right)^{2} F_{46}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= -\frac{y \left(F_{62}\! \left(x , 1\right)-F_{62}\! \left(x , y\right)\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Isolated" and has 32 rules.
Found on April 20, 2021.Finding the specification took 1163 seconds.
Copy 32 equations to clipboard:
\(\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_{19}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \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_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{20}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{3}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{10}\! \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_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{25}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \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_{19}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{19}\! \left(x \right) F_{29}\! \left(x \right)\\
\end{align*}\)