PermPal Database

Av(1234, 2431, 3412)

Generating Function

- \frac{3 x^{6} - 6 x^{5} + 21 x^{4} - 22 x^{3} + 16 x^{2} - 6 x + 1}{{(- 1 + x)}^{7}}

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Counting Sequence

1, 1, 2, 6, 21, 71, 213, 561, 1317, 2809, 5536, 10220, 17865, 29823, 47867, ...

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Implicit Equation for the Generating Function

{(- 1 + x)}^{7} F (x) + 3 x^{6} - 6 x^{5} + 21 x^{4} - 22 x^{3} + 16 x^{2} - 6 x + 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (4) = 21

a (5) = 71

a (6) = 213

a (n) = 1 + \frac{7}{720} n^{6} - \frac{1}{16} n^{5} + \frac{35}{144} n^{4} - \frac{7}{16} n^{3} + \frac{269}{360} n^{2} - \frac{1}{2} n, n \geq 7

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Explicit Closed Form

1 + \frac{7}{720} n^{6} - \frac{1}{16} n^{5} + \frac{35}{144} n^{4} - \frac{7}{16} n^{3} + \frac{269}{360} n^{2} - \frac{1}{2} n

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This specification was found using the strategy pack "Point Placements" and has 101 rules.

Found on January 18, 2022.

Finding the specification took 11 seconds.

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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_{33} (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_{13} (x) \\ F_{10} (x) & = F_{1} (x) + F_{11} (x) \\ F_{11} (x) & = F_{12} (x) \\ F_{12} (x) & = F_{10} (x) F_{4} (x) \\ F_{13} (x) & = F_{14} (x) + F_{7} (x) \\ F_{14} (x) & = F_{15} (x) + F_{16} (x) + F_{28} (x) \\ F_{15} (x) & = 0 \\ F_{16} (x) & = F_{17} (x) F_{4} (x) \\ F_{17} (x) & = F_{18} (x) + F_{22} (x) \\ F_{18} (x) & = F_{11} (x) + F_{19} (x) \\ F_{19} (x) & = F_{20} (x) \\ F_{20} (x) & = F_{21} (x) F_{4} (x) \\ F_{21} (x) & = F_{11} (x) \\ F_{22} (x) & = F_{14} (x) + F_{23} (x) \\ F_{23} (x) & = F_{24} (x) \\ F_{24} (x) & = F_{25} (x) F_{4} (x) \\ F_{25} (x) & = F_{26} (x) + F_{27} (x) \\ F_{26} (x) & = F_{19} (x) \\ F_{27} (x) & = F_{23} (x) \\ F_{28} (x) & = F_{29} (x) F_{4} (x) \\ F_{29} (x) & = F_{11} (x) + F_{30} (x) \\ F_{30} (x) & = F_{31} (x) \\ F_{31} (x) & = F_{32} (x) F_{4} (x) \\ F_{32} (x) & = F_{11} (x) + F_{30} (x) \\ F_{33} (x) & = F_{2} (x) + F_{34} (x) \\ F_{34} (x) & = F_{15} (x) + F_{35} (x) + F_{95} (x) \\ F_{35} (x) & = F_{36} (x) F_{4} (x) \\ F_{36} (x) & = F_{37} (x) + F_{51} (x) \\ F_{37} (x) & = F_{38} (x) + F_{7} (x) \\ F_{38} (x) & = F_{15} (x) + F_{16} (x) + F_{39} (x) \\ F_{39} (x) & = F_{4} (x) F_{40} (x) \\ F_{40} (x) & = F_{41} (x) + F_{45} (x) \\ F_{41} (x) & = F_{11} (x) + F_{42} (x) \\ F_{42} (x) & = F_{15} (x) + F_{28} (x) + F_{43} (x) \\ F_{43} (x) & = F_{4} (x) F_{44} (x) \\ F_{44} (x) & = F_{11} (x) + F_{42} (x) \\ F_{45} (x) & = F_{46} (x) + F_{47} (x) \\ F_{46} (x) & = F_{31} (x) \\ F_{47} (x) & = F_{48} (x) \\ F_{48} (x) & = F_{4} (x) F_{49} (x) \\ F_{49} (x) & = F_{19} (x) + F_{50} (x) \\ F_{50} (x) & = F_{48} (x) \\ F_{51} (x) & = F_{34} (x) + F_{52} (x) \\ F_{52} (x) & = F_{15} (x) + F_{53} (x) + F_{88} (x) + F_{94} (x) \\ F_{53} (x) & = F_{4} (x) F_{54} (x) \\ F_{54} (x) & = F_{55} (x) + F_{65} (x) \\ F_{55} (x) & = F_{38} (x) + F_{56} (x) \\ F_{56} (x) & = F_{15} (x) + F_{24} (x) + F_{57} (x) + F_{64} (x) \\ F_{57} (x) & = F_{4} (x) F_{58} (x) \\ F_{58} (x) & = F_{59} (x) + F_{63} (x) \\ F_{59} (x) & = F_{42} (x) + F_{60} (x) \\ F_{60} (x) & = F_{61} (x) \\ F_{61} (x) & = F_{4} (x) F_{62} (x) \\ F_{62} (x) & = F_{19} (x) + F_{60} (x) \\ F_{63} (x) & = F_{47} (x) \\ F_{64} (x) & = 0 \\ F_{65} (x) & = F_{52} (x) + F_{66} (x) \\ F_{66} (x) & = F_{15} (x) + F_{67} (x) + F_{80} (x) + F_{86} (x) + F_{87} (x) \\ F_{67} (x) & = F_{4} (x) F_{68} (x) \\ F_{68} (x) & = F_{69} (x) + F_{74} (x) \\ F_{69} (x) & = F_{56} (x) + F_{70} (x) \\ F_{70} (x) & = F_{71} (x) \\ F_{71} (x) & = F_{4} (x) F_{72} (x) \\ F_{72} (x) & = F_{73} (x) \\ F_{73} (x) & = F_{60} (x) \\ F_{74} (x) & = F_{66} (x) + F_{75} (x) \\ F_{75} (x) & = F_{76} (x) \\ F_{76} (x) & = F_{4} (x) F_{77} (x) \\ F_{77} (x) & = F_{78} (x) + F_{79} (x) \\ F_{78} (x) & = F_{70} (x) \\ F_{79} (x) & = F_{75} (x) \\ F_{80} (x) & = F_{4} (x) F_{81} (x) \\ F_{81} (x) & = F_{73} (x) + F_{82} (x) \\ F_{82} (x) & = F_{83} (x) \\ F_{83} (x) & = F_{84} (x) \\ F_{84} (x) & = F_{4} (x) F_{85} (x) \\ F_{85} (x) & = F_{23} (x) + F_{83} (x) \\ F_{86} (x) & = 0 \\ F_{87} (x) & = 0 \\ F_{88} (x) & = F_{4} (x) F_{89} (x) \\ F_{89} (x) & = F_{59} (x) + F_{90} (x) \\ F_{90} (x) & = F_{83} (x) + F_{91} (x) \\ F_{91} (x) & = F_{92} (x) \\ F_{92} (x) & = F_{4} (x) F_{93} (x) \\ F_{93} (x) & = F_{14} (x) + F_{91} (x) \\ F_{94} (x) & = 0 \\ F_{95} (x) & = F_{4} (x) F_{96} (x) \\ F_{96} (x) & = F_{41} (x) + F_{97} (x) \\ F_{97} (x) & = F_{91} (x) + F_{98} (x) \\ F_{98} (x) & = F_{99} (x) \\ F_{99} (x) & = F_{100} (x) F_{4} (x) \\ F_{100} (x) & = F_{7} (x) + F_{98} (x) \end{aligned}

Av(1234, 2431, 3412)

This specification was found using the strategy pack "Point Placements" and has 101 rules.

Proof Tree

System of Equations