Av(1243, 2314, 2341, 3412)
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Generating Function
\(\displaystyle \frac{2 x^{8}-6 x^{7}+35 x^{6}-78 x^{5}+97 x^{4}-75 x^{3}+35 x^{2}-9 x +1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{6}}\)
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Counting Sequence
1, 1, 2, 6, 20, 63, 180, 470, 1144, 2647, 5916, 12926, 27844, 59463, 126324, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \! \left(x \right)-2 x^{8}+6 x^{7}-35 x^{6}+78 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 180\)
\(\displaystyle a \! \left(7\right) = 470\)
\(\displaystyle a \! \left(8\right) = 1144\)
\(\displaystyle a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{n \left(n^{4}-20 n^{3}+105 n^{2}-190 n +224\right)}{60}, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{n^{5}}{60}-\frac{n^{4}}{6}+\frac{5 n^{3}}{12}-\frac{4 n^{2}}{3}+\frac{n}{15}+2^{-1+n} n +2^{-1+n} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point And Row Placements" and has 141 rules.

Found on July 23, 2021.

Finding the specification took 11 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_{17}\! \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_{126}\! \left(x \right)+F_{6}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{6}\! \left(x \right) &= 0\\ F_{7}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{43}\! \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_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{17}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{17}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{17}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{17}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{17}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{6}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{17}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{17}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{17}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{69}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{17}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{17}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{17}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{17}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{17}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{17}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{91}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{17}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{17}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{17}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{105}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{111}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{112}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{107}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{115}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{116}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{17}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{10}\! \left(x \right) F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{140}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{136}\! \left(x \right) F_{17}\! \left(x \right)\\ \end{align*}\)