Av(1234, 1243, 1324, 1342, 1432)
Generating Function
\(\displaystyle \frac{x^{4}+x^{3}-4 x^{2}+4 x -1}{4 x^{3}-7 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 61, 196, 629, 2017, 6466, 20727, 66441, 212980, 682721, 2188509, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-4 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+x^{4}+x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = 4 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = 4 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-4543 \,\mathrm{I}-418 \sqrt{59}\right) \sqrt{3}+1254 \,\mathrm{I} \sqrt{59}+4543\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+71390+\left(\left(-1003 \,\mathrm{I}+20 \sqrt{59}\right) \sqrt{3}+60 \,\mathrm{I} \sqrt{59}-1003\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}-18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}-\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{685344}\\+\\\frac{\left(\left(\left(4543 \,\mathrm{I}-418 \sqrt{59}\right) \sqrt{3}-1254 \,\mathrm{I} \sqrt{59}+4543\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+71390+\left(\left(1003 \,\mathrm{I}+20 \sqrt{59}\right) \sqrt{3}-60 \,\mathrm{I} \sqrt{59}-1003\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}+18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}+\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{685344}\\+\\\frac{19 \left(-\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{7}{12}-\frac{71 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{1452}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{59}\, \sqrt{3}}{242}\right)^{-n} \left(\left(-\frac{10 \sqrt{59}\, \sqrt{3}}{209}+\frac{1003}{418}\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, \sqrt{3}-\frac{413 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{38}+\frac{3245}{38}\right)}{15576} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 107 rules.
Found on January 18, 2022.Finding the specification took 3 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_{14}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \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_{29}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{16}\! \left(x \right)+F_{31}\! \left(x \right)+F_{75}\! \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_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 0\\
F_{44}\! \left(x \right) &= 0\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{16}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{63}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \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_{66}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{16}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{75}\! \left(x \right)+F_{86}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{99}\! \left(x \right)\\
\end{align*}\)