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(Re) Permutations Involving Relatively-Primality
Posted:
Dec 28, 2002 2:59 AM
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This is way past the thread's use-by date, but I finally got around to this
part of my back-log, and thought Leroy (and others?) might still be faintly
interested. Sorry if followups have already covered it.
Not that I'm sure I'm not sending this into cyber-vacuum anyway -- traffic
seems to have dropped almost to zero! I mean I know it's holiday time
and all, but I don't recall Usenet being QUITE so bereft in previous years.
Anyway, Leroy Quet wrote:
> What is the number of permutations of the first m positive integers where
> GCD(a(k-1), a(k)) = 1
Quite a fun counting problem, in fact.
> With a brute-force counting Mathematica program...
> 1, 2, 6, 12, 72, 72, 864, 1728,...
I managed to varify these by hand, (NOT by listing them all, natch!)
It seems to extend to:
1, 2, 6, 12, 72, 72, 864, 1728, 8928, ... so that...
> Noteworthy: For the terms given, anyway, each term is a multiple of
> the term before it.
...this does not in fact continue to apply. I'm slightly glad it doesn't,
because I couldn't think of any likely semi-reason why it should!
> variation of the question: What if a(m) must also be relatively-prime to a(1) ?
In fact this version is somewhat easier to tackle, by hand at least.
My tentative numbers come out as:
1 2 6 8 60 24 504 576 7776 5760 ...
Interestingly the sequence is no longer monotonic.
Hopefully someone else will take up the baton and run a little further with it?
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Bill Taylor W.Taylor@math.canterbury.ac.nz
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Does a political barge horse have to tow the party line?
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