Av(1243, 1342, 2314, 3124)
Generating Function
\(\displaystyle -\frac{\left(-3 x^{2}+\left(x^{2}-x +1\right) \sqrt{1-4 x}+3 x -1\right) \left(x -1\right)^{2}}{2 \left(x^{3}-x +1\right) x^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 225, 764, 2631, 9181, 32402, 115453, 414754, 1500578, 5463000, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-x +1\right) x^{2} F \left(x
\right)^{2}-\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 129 rules.
Found on July 23, 2021.Finding the specification took 10 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_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{126}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{22}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{22}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \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_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \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_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
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_{22}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\
F_{23}\! \left(x , y\right) &= y x\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{22}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{22}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x , 1\right)\\
F_{38}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x , y\right)+F_{44}\! \left(x , y\right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{3}\! \left(x \right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{38}\! \left(x , y\right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x , 1\right)\\
F_{46}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x \right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\
F_{50}\! \left(x \right) &= F_{22}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x , 1\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= -\frac{-y F_{56}\! \left(x , y\right)+F_{56}\! \left(x , 1\right)}{-1+y}\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x \right)+F_{70}\! \left(x , y\right)\\
F_{58}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{22}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 0\\
F_{65}\! \left(x \right) &= F_{22}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{70}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{64}\! \left(x \right)+F_{72}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{59}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{75}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{76}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{64}\! \left(x \right)+F_{81}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{80}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= 2 F_{64}\! \left(x \right)+F_{86}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{87}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{89}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{41}\! \left(x , y\right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{94}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= -\frac{-y F_{47}\! \left(x , y\right)+F_{47}\! \left(x , 1\right)}{-1+y}\\
F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= -\frac{-y F_{55}\! \left(x , y\right)+F_{55}\! \left(x , 1\right)}{-1+y}\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{100}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)+F_{124}\! \left(x , y\right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{27}\! \left(x \right) F_{3}\! \left(x \right) F_{41}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{122}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)+F_{11}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{112}\! \left(x , y\right)+F_{64}\! \left(x \right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{16}\! \left(x \right)\\
F_{111}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{118}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{121}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\
F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{27} \left(x \right)^{2} F_{22}\! \left(x \right) F_{41}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right) F_{22}\! \left(x \right)\\
F_{125}\! \left(x , y\right) &= -\frac{-y F_{38}\! \left(x , y\right)+F_{38}\! \left(x , 1\right)}{-1+y}\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x , 1\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)\\
F_{128}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{56}\! \left(x , y\right)\\
\end{align*}\)