
(Re) Permutations Involving RelativelyPrimality
Posted:
Dec 28, 2002 2:59 AM


This is way past the thread's useby date, but I finally got around to this part of my backlog, 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 cybervacuum 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(k1), a(k)) = 1
Quite a fun counting problem, in fact.
> With a bruteforce 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 semireason why it should!
> variation of the question: What if a(m) must also be relativelyprime 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?
 Bill Taylor W.Taylor@math.canterbury.ac.nz  Does a political barge horse have to tow the party line? 

