Av(1243, 1324, 2314, 2341, 4123)
Generating Function
\(\displaystyle \frac{-\left(x^{2}+x -1\right) \left(x -1\right)^{4} \sqrt{1-4 x}-2 x^{8}+2 x^{7}+5 x^{6}-3 x^{5}-x^{4}+6 x^{3}-9 x^{2}+5 x -1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 162, 485, 1526, 5018, 17041, 59163, 208586, 743771, 2675764, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(x^{2}+x -1\right) \left(2 x^{8}-2 x^{7}-5 x^{6}+3 x^{5}+x^{4}-6 x^{3}+9 x^{2}-5 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{15}-2 x^{14}-4 x^{13}+9 x^{12}-2 x^{11}-2 x^{10}+18 x^{9}-52 x^{8}+64 x^{7}-11 x^{6}-85 x^{5}+134 x^{4}-101 x^{3}+43 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(7\right) = 485\)
\(\displaystyle a \! \left(8\right) = 1526\)
\(\displaystyle a \! \left(9\right) = 5018\)
\(\displaystyle a \! \left(10\right) = 17041\)
\(\displaystyle a \! \left(11\right) = 59163\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{6+n}+\frac{\left(7 n +10\right) a \! \left(1+n \right)}{6+n}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{6+n}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{6+n}-\frac{n^{4}-12 n^{3}-11 n^{2}+120 n +92}{2 \left(6+n \right)}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(7\right) = 485\)
\(\displaystyle a \! \left(8\right) = 1526\)
\(\displaystyle a \! \left(9\right) = 5018\)
\(\displaystyle a \! \left(10\right) = 17041\)
\(\displaystyle a \! \left(11\right) = 59163\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{6+n}+\frac{\left(7 n +10\right) a \! \left(1+n \right)}{6+n}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{6+n}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{6+n}-\frac{n^{4}-12 n^{3}-11 n^{2}+120 n +92}{2 \left(6+n \right)}, \quad n \geq 12\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 101 rules.
Found on July 23, 2021.Finding the specification took 2 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \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_{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_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= x^{2}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 0\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{36}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x , 1\right)\\
F_{76}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x , y\right)+F_{77}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)+F_{87}\! \left(x , y\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_{1}\! \left(x \right)+F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{80}\! \left(x , y\right) F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= y x\\
F_{84}\! \left(x , y\right) &= F_{32}\! \left(x \right)+F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{83}\! \left(x , y\right) F_{84}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{32}\! \left(x \right)+F_{89}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{90}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{83}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{49}\! \left(x \right)+F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= 2 F_{36}\! \left(x \right)+F_{95}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{83}\! \left(x , y\right) F_{93}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= \frac{F_{76}\! \left(x , y\right) y -F_{76}\! \left(x , 1\right)}{-1+y}\\
F_{100}\! \left(x , y\right) &= F_{76}\! \left(x , y\right) F_{83}\! \left(x , y\right)\\
\end{align*}\)