Av(1234, 1342, 1432, 4123, 4213)
Generating Function
\(\displaystyle -\frac{2 x^{9}+6 x^{8}+5 x^{7}+3 x^{6}+4 x^{5}+7 x^{4}+2 x^{3}-2 x +1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 51, 127, 302, 686, 1505, 3222, 6766, 13993, 28591, 57841, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{3}+x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{9}+6 x^{8}+5 x^{7}+3 x^{6}+4 x^{5}+7 x^{4}+2 x^{3}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 127\)
\(\displaystyle a \! \left(7\right) = 302\)
\(\displaystyle a \! \left(8\right) = 686\)
\(\displaystyle a \! \left(9\right) = 1505\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}+\frac{28}{3}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 127\)
\(\displaystyle a \! \left(7\right) = 302\)
\(\displaystyle a \! \left(8\right) = 686\)
\(\displaystyle a \! \left(9\right) = 1505\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}+\frac{28}{3}, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(-561 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(-\frac{315}{187}+n \right) \sqrt{11}+1001 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{163}{91}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-132 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n +\frac{3}{22}\right) \sqrt{11}+374 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{8}{17}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+2992 n -13552\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5808}\\+\\\frac{\left(\left(561 \left(-\frac{315}{187}+n \right) \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{11}-1001 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{163}{91}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(132 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n +\frac{3}{22}\right) \sqrt{11}-374 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{8}{17}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+2992 n -13552\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5808}\\+7+\\\frac{\left(\left(\left(374 n -630\right) \sqrt{3}\, \sqrt{11}-2002 n +3586\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-88 n -12\right) \sqrt{3}\, \sqrt{11}+748 n -352\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+2992 n -13552\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{5808} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 97 rules.
Found on January 18, 2022.Finding the specification took 1 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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \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_{11}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{31}\! \left(x \right) &= 0\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= x^{2}\\
F_{45}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= x^{2}\\
F_{71}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{66}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{87}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{49}\! \left(x \right)\\
\end{align*}\)