Av(1234, 1243, 1324, 3412)
Generating Function
\(\displaystyle \frac{2 x^{7}-3 x^{6}+16 x^{5}-31 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(-1+x \right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 180, 468, 1126, 2553, 5542, 11666, 24048, 48899, 98592, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)-2 x^{7}+3 x^{6}-16 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 180\)
\(\displaystyle a \! \left(7\right) = 468\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-19 n^{3}+66 n^{2}-154 n +60\right)}{60}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 180\)
\(\displaystyle a \! \left(7\right) = 468\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-19 n^{3}+66 n^{2}-154 n +60\right)}{60}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -6+6 \,2^{n}+\frac{n^{5}}{60}-\frac{n^{4}}{4}+\frac{7 n^{3}}{12}-\frac{13 n^{2}}{4}-\frac{21 n}{10} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Insertion Point Placements" and has 168 rules.
Found on July 23, 2021.Finding the specification took 17 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
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_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
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_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{16} \left(x \right)^{2} F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{76}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{8}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{8}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{8}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{8}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{8}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{48}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{8}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{101}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{102}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{25}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{110}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{105}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{8}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{6}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{106}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{140}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{133}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{143}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{142}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{142}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{150}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{14}\! \left(x \right) F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{155}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{158}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{159}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{158}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{162}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{14}\! \left(x \right) F_{164}\! \left(x \right)\\
\end{align*}\)