Av(1234, 1324, 1342, 1423, 3214)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{7}+x^{6}+x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 162, 471, 1376, 4023, 11758, 34360, 100412, 293450, 857604, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+x^{6}+x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)-4 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)-4 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +5}}{163223773}-\frac{4913210 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +5}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{163223773}-\frac{12379582 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +4}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{163223773}-\frac{17539116 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +3}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{163223773}-\frac{12857801 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +2}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{163223773}+\frac{8436448 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{163223773}+\frac{8632653 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n -1}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{163223773}+\frac{16507266 \mathit{RootOf} \left(Z^{7}+Z^{6}+Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n}}{163223773}\)
This specification was found using the strategy pack "Point Placements" and has 61 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 61 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_{14}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
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_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{45}\! \left(x \right)\\
\end{align*}\)