PermPal Database

Av(1342, 3214)

Generating Function

- \frac{(x - 1) (3 x - 1) (x^{2} - 3 x + 1)}{2 x^{5} - 10 x^{4} + 25 x^{3} - 22 x^{2} + 8 x - 1}

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Counting Sequence

1, 1, 2, 6, 22, 86, 336, 1290, 4870, 18164, 67234, 247786, 911120, 3346618, 12286942, ...

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Implicit Equation for the Generating Function

(2 x^{5} - 10 x^{4} + 25 x^{3} - 22 x^{2} + 8 x - 1) F (x) + (x - 1) (3 x - 1) (x^{2} - 3 x + 1) = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 22

a (n + 5) = 2 a (n) - 10 a (n + 1) + 25 a (n + 2) - 22 a (n + 3) + 8 a (n + 4), n \geq 5

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Explicit Closed Form

- \frac{9310 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 1)}^{- n + 3}}{17539} - \frac{9310 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 2)}^{- n + 3}}{17539} - \frac{9310 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 3)}^{- n + 3}}{17539} - \frac{9310 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 4)}^{- n + 3}}{17539} - \frac{9310 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 5)}^{- n + 3}}{17539} + \frac{41357 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 1)}^{- n + 2}}{17539} + \frac{41357 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 2)}^{- n + 2}}{17539} + \frac{41357 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 3)}^{- n + 2}}{17539} + \frac{41357 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 4)}^{- n + 2}}{17539} + \frac{41357 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 5)}^{- n + 2}}{17539} - \frac{93738 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 1)}^{- n + 1}}{17539} - \frac{93738 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 2)}^{- n + 1}}{17539} - \frac{93738 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 3)}^{- n + 1}}{17539} - \frac{93738 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 4)}^{- n + 1}}{17539} - \frac{93738 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 5)}^{- n + 1}}{17539} - \frac{8207 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 1)}^{- n - 1}}{17539} - \frac{8207 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 2)}^{- n - 1}}{17539} - \frac{8207 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 3)}^{- n - 1}}{17539} - \frac{8207 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 4)}^{- n - 1}}{17539} - \frac{8207 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 5)}^{- n - 1}}{17539} + \frac{55448 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 1)}^{- n}}{17539} + \frac{55448 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 2)}^{- n}}{17539} + \frac{55448 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 3)}^{- n}}{17539} + \frac{55448 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 4)}^{- n}}{17539} + \frac{55448 RootOf {(2 Z^{5} - 10 Z^{4} + 25 Z^{3} - 22 Z^{2} + 8 Z - 1, index = 5)}^{- n}}{17539}

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Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point $(i, j)$ represents how many permutations have value $j$ at index $i$ (darker = more).

This specification was found using the strategy pack "Regular Insertion Encoding Right" and has 70 rules.

Found on April 28, 2021.

Finding the specification took 3 seconds.

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+ Expand

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{4} (x) F_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{15} (x) + F_{6} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{9} (x) \\ F_{9} (x) & = F_{10} (x) + F_{6} (x) \\ F_{10} (x) & = F_{11} (x) + F_{7} (x) \\ F_{11} (x) & = F_{12} (x) \\ F_{12} (x) & = F_{13} (x) F_{4} (x) \\ F_{13} (x) & = F_{14} (x) \\ F_{14} (x) & = F_{11} (x) + F_{7} (x) \\ F_{15} (x) & = F_{16} (x) + F_{2} (x) \\ F_{16} (x) & = F_{17} (x) + F_{18} (x) + F_{56} (x) \\ F_{17} (x) & = 0 \\ F_{18} (x) & = F_{19} (x) F_{4} (x) \\ F_{19} (x) & = F_{20} (x) + F_{31} (x) \\ F_{20} (x) & = F_{21} (x) + F_{7} (x) \\ F_{21} (x) & = F_{17} (x) + F_{22} (x) + F_{24} (x) \\ F_{22} (x) & = F_{23} (x) F_{4} (x) \\ F_{23} (x) & = F_{20} (x) \\ F_{24} (x) & = F_{25} (x) F_{4} (x) \\ F_{25} (x) & = F_{10} (x) + F_{26} (x) \\ F_{26} (x) & = F_{21} (x) + F_{27} (x) \\ F_{27} (x) & = F_{28} (x) \\ F_{28} (x) & = F_{29} (x) F_{4} (x) \\ F_{29} (x) & = F_{30} (x) \\ F_{30} (x) & = F_{11} (x) + F_{27} (x) \\ F_{31} (x) & = F_{32} (x) + F_{41} (x) \\ F_{32} (x) & = F_{33} (x) \\ F_{33} (x) & = F_{34} (x) F_{4} (x) \\ F_{34} (x) & = F_{35} (x) + F_{36} (x) \\ F_{35} (x) & = F_{2} (x) + F_{32} (x) \\ F_{36} (x) & = F_{32} (x) + F_{37} (x) \\ F_{37} (x) & = F_{38} (x) \\ F_{38} (x) & = F_{39} (x) F_{4} (x) \\ F_{39} (x) & = F_{40} (x) \\ F_{40} (x) & = F_{32} (x) + F_{37} (x) \\ F_{41} (x) & = 2 F_{17} (x) + F_{42} (x) + F_{44} (x) \\ F_{42} (x) & = F_{4} (x) F_{43} (x) \\ F_{43} (x) & = F_{31} (x) \\ F_{44} (x) & = F_{4} (x) F_{45} (x) \\ F_{45} (x) & = F_{46} (x) + F_{51} (x) \\ F_{46} (x) & = F_{16} (x) + F_{47} (x) \\ F_{47} (x) & = F_{48} (x) \\ F_{48} (x) & = F_{4} (x) F_{49} (x) \\ F_{49} (x) & = F_{50} (x) \\ F_{50} (x) & = F_{32} (x) + F_{47} (x) \\ F_{51} (x) & = F_{41} (x) + F_{52} (x) \\ F_{52} (x) & = F_{53} (x) \\ F_{53} (x) & = F_{4} (x) F_{54} (x) \\ F_{54} (x) & = F_{55} (x) \\ F_{55} (x) & = F_{37} (x) + F_{52} (x) \\ F_{56} (x) & = F_{4} (x) F_{57} (x) \\ F_{57} (x) & = F_{15} (x) + F_{58} (x) \\ F_{58} (x) & = F_{16} (x) + F_{59} (x) \\ F_{59} (x) & = 2 F_{17} (x) + F_{48} (x) + F_{60} (x) \\ F_{60} (x) & = F_{4} (x) F_{61} (x) \\ F_{61} (x) & = F_{62} (x) + F_{66} (x) \\ F_{62} (x) & = F_{11} (x) + F_{63} (x) \\ F_{63} (x) & = F_{64} (x) \\ F_{64} (x) & = F_{4} (x) F_{65} (x) \\ F_{65} (x) & = F_{62} (x) \\ F_{66} (x) & = F_{37} (x) + F_{67} (x) \\ F_{67} (x) & = F_{68} (x) \\ F_{68} (x) & = F_{4} (x) F_{69} (x) \\ F_{69} (x) & = F_{66} (x) \end{aligned}

Av(1342, 3214)

This specification was found using the strategy pack "Regular Insertion Encoding Right" and has 70 rules.

Proof Tree

System of Equations