Martin Rubey <[EMAIL PROTECTED]> writes:
> > > Similarly, if $G$ is a species, the longest
> > > cycle of $G[\sigma]$ can only be shorter than the the longest cycle of
> > > $\sigma$. (Check this...!)
> >
> > You fell into the same trap as me. Diff revision 158 and 157, I've just
> > commited a small docfix. It is right there. Or see at BLL chapter 1.2 after
> > definition 5. This is a counter example.
>
> I don't see what this has to do with
>
> Z_F(x_1,x_2,\ldots) \ne \sum_{n\geq0}C(S_n,F[n];x_1,x_2,\ldots,x_n).
>
> but I see that BLL indeed provides a counter example. However,
>
> > The cycles of G[\sigma] are "potentially" as long |G[n]|, right?
>
> seems to be a very crude upper bound. But, of course, the sum of the cycle
> lengths of G[\sigma] must be |G[n]|, not n... Stupid me.
I just realised: the length of the longest cycle is the lcm of the cycle
lengths of \sigma.
Proof:
Let s be a structure on [n] and let \pi be a permutation of [n].
We want to know the minimal positive number greater than one with %
$\pi^o s = s$, which is just the order of $\pi$. This is just the least common
multiple of the cycle lengths of \pi.
Martin
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