Av(1234, 1432, 2143, 2413, 3214)
View Raw Data
Generating Function
\(\displaystyle -\frac{1}{\left(x +1\right) \left(x^{6}+5 x^{5}+4 x^{4}+4 x^{3}-x^{2}+2 x -1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 48, 116, 294, 773, 2008, 5143, 13173, 33914, 87417, 224998, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right) \left(x^{6}+5 x^{5}+4 x^{4}+4 x^{3}-x^{2}+2 x -1\right) F \! \left(x \right)+1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 48\)
\(\displaystyle a \! \left(6\right) = 116\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+6 a \! \left(n +1\right)+9 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +5}}{228980896}+\frac{2577157 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +5}}{228980896}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +4}}{686942688}+\frac{40294879 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +4}}{686942688}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +3}}{57245224}+\frac{3862385 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +3}}{57245224}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +2}}{171735672}+\frac{20702951 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +2}}{171735672}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +1}}{686942688}+\frac{86665025 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n -1}}{686942688}+\frac{34553597 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n -1}}{686942688}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n}}{343471344}+\frac{\left(-1\right)^{-n}}{8}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n}}{343471344}+\frac{69029933 \mathit{RootOf} \left(Z^{6}+5 Z^{5}+4 Z^{4}+4 Z^{3}-Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n}}{343471344}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 75 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 75 equations to clipboard:
\(\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_{25}\! \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_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= x^{2}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \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_{35}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \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_{52}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{41}\! \left(x \right)\\ \end{align*}\)