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COMMENT
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The number of functions in a finite set {1,..,n} for which the sequence of composition powers ends in a fixed point gave terms of the sequence A000272(n-1)=(n+1)^(n-1).
This is to be seen as a conjecture, and the sequence ending with a lenght 2 cycle does not seem to have such an easy expression.
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EXAMPLE
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Any transposition (or disjoint combination) is one element to be counted.
When n=2, there is only one, and a(2)=1. When n=3, there are only 3 transpositions, but there are other 6 elements, for instance
f:{1,2,3}->{2,1,1} gives fof:{1,2,3}->{1,2,2} and fofof=f (cycle 2),
(the others are similar), thus giving a(3)=9.
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