Av(1234, 1342, 2431, 3142, 3214)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(2 x^{6}+x^{5}-7 x^{4}+3 x^{3}-4 x^{2}+3 x -1\right)}{\left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 50, 113, 252, 565, 1255, 2762, 6071, 13351, 29343, 64450, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(x +1\right) \left(2 x^{6}+x^{5}-7 x^{4}+3 x^{3}-4 x^{2}+3 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 113\)
\(\displaystyle a \! \left(7\right) = 252\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(4+n \right)+6 n +19, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{52701 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +1}}{51569}\\+\\\frac{299050 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +2}}{360983}\\+\\\frac{261417 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3}}{360983}\\+\\\frac{93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{360983}\\+\\\frac{\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}-281643 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-187762 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+275026\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +1}}{360983}\\+\\\frac{\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-281643 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+111288\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +2}}{360983}\\+\\\frac{\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3}}{360983}\\+\\\frac{\left(\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+301869 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+60678\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+60678 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+315478\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +1}}{360983}\\+\\\frac{\left(\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +2}}{360983}\\+\\\frac{\left(\left(\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-171966 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-200420\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +1}}{360983}\\+\\\frac{124492 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{360983}\\+\\\frac{\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{4}-281643 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}-187762 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+30611\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{360983}\\+\\\frac{\left(\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}+\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+301869 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+60678\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+301869 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+248440 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+40452\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+60678 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+40452 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+50837\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{360983}\\+\\\frac{\left(\left(\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-322095 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-232644\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-232644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-515898\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-232644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-515898\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-171966 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}-515898 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-293095\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{360983}\\+\\\frac{\left(\left(\left(\left(93881 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+20226\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+171966 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+200420\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(\left(20226 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+171966\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+171966 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+200420\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(171966 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+200420\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+200420 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+308165\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{360983}\\-\frac{6 n}{7}-\frac{79}{49} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 56 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{17}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \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_{20}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ 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_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{44}\! \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_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right)\\ \end{align*}\)