Landau's function gives the maximum cycle length of a permutation.
See.,e.g.,
http://mathworld.wolfram.com/LandausFunction.html.
Bill Dunlap
TIBCO Software
wdunlap tibco.com
On Thu, Dec 7, 2017 at 9:34 PM, Eric Berger <ericjber...@gmail.com> wrote:
> Hi Boris,
> Do a search on "the order of elements of the symmetric group". (This search
> will also turn up homework questions and solutions.) You will understand
> why you are seeing this once you understand how a permutation is decomposed
> into cycles and how the order relates to a partition of n (n=10 in your
> case).
>
> Enjoy!
> Eric
>
>
> On Fri, Dec 8, 2017 at 6:39 AM, Boris Steipe <boris.ste...@utoronto.ca>
> wrote:
>
> > I have noticed that when I iterate permutations of short vectors with the
> > same seed, the cycle lengths are much shorter than I would expect by
> > chance. For example:
> >
> > X <- 1:10
> > Xorig <- X
> > start <- 112358
> > N <- 10
> >
> > for (i in 1:N) {
> > seed <- start + i
> > for (j in 1:1000) { # Maximum cycle length to consider
> > set.seed(seed) # Re-seed RNG to same initial state
> > X <- sample(X) # Permute X and iterate
> > if (all(X == Xorig)) {
> > cat(sprintf("Seed:\t%d\tCycle: %d\n", seed, j))
> > break()
> > }
> > }
> > }
> >
> > Seed: 112359 Cycle: 14
> > Seed: 112360 Cycle: 14
> > Seed: 112361 Cycle: 8
> > Seed: 112362 Cycle: 14
> > Seed: 112363 Cycle: 8
> > Seed: 112364 Cycle: 10
> > Seed: 112365 Cycle: 10
> > Seed: 112366 Cycle: 10
> > Seed: 112367 Cycle: 9
> > Seed: 112368 Cycle: 12
> >
> > I understand that I am performing the same permutation operation over and
> > over again - but I don't see why that would lead to such a short cycle
> (in
> > fact the cycle for the first 100,000 seeds is never longer than 30). Does
> > this have a straightforward explanation?
> >
> >
> > Thanks!
> > Boris
> >
> > ______________________________________________
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> > PLEASE do read the posting guide http://www.R-project.org/
> > posting-guide.html
> > and provide commented, minimal, self-contained, reproducible code.
> >
>
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>
> ______________________________________________
> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see
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>
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