Av(1234, 1342, 2314, 2341)
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Generating Function
\(\displaystyle -\frac{\left(-1+2 x +\sqrt{-4 x +1}\right) \left(-1+2 x \right)}{2 x^{2} \left(-1+x \right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 233, 805, 2807, 9879, 35073, 125513, 452389, 1641029, 5986994, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(-1+x \right)^{2} F \left(x \right)^{2}+\left(-1+x \right) \left(-1+2 x \right)^{2} F \! \left(x \right)+\left(-1+2 x \right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(25+7 n \right) a \! \left(2+n \right)}{5+n}-\frac{2 \left(16+7 n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 137 rules.

Found on July 23, 2021.

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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{136}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= y x\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= \frac{F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{64}\! \left(x , y\right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= 0\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{11}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{11}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{11}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{64}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{67}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{70}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{77}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{21}\! \left(x \right)+F_{76}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{75}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{85}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{84}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{82}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{91}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{21}\! \left(x \right)+F_{90}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{97}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{32}\! \left(x \right)+F_{96}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{99}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{105}\! \left(x , y\right)+F_{107}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{106}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{114}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{110}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{111}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{112}\! \left(x , y\right)\\ F_{112}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\ F_{113}\! \left(x , y\right) &= F_{109}\! \left(x , y\right) F_{11}\! \left(x \right)\\ F_{114}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{117}\! \left(x , y\right)\\ F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= 3 F_{25}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{120}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{118}\! \left(x , y\right)\\ F_{121}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{122}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{130}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= 2 F_{25}\! \left(x \right)+F_{125}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{126}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{127}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{128}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{129}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)\\ F_{130}\! \left(x , y\right) &= 3 F_{25}\! \left(x \right)+F_{131}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\ F_{131}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{132}\! \left(x , y\right)\\ F_{132}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{133}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{134}\! \left(x , y\right)\\ F_{134}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)\\ F_{136}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\ \end{align*}\)