Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> > I think the best one can hope for is to sum over cycletypes (i.e.,
> > partitions) instead of permutations.
>
> But that is exactly how the cycle index series looks like.
Yes. What are you hoping for?
> Let H = F \square G, then I want the coefficient of x^k of Z_H (k the cycle
> type) in terms of the coefficients of some other x^i in Z_F and Z_G.
> > The formula for the cycle index series now becomes
> > \begin{equation*}
> > Z_F \square Z_G = \sum_{n_1+2n_2+\dots}
> > \fix(F\square G)[n_1, n_2, \dots]
> > \frac{x_1^{n_1}x_2^{n_2}\dots}
> > {1^{n_1}n_1! 2^{n_2}n_2!\dots}.
> > \end{equation*}
>
> This Formula is not at all helpful. And it is true by definition of the
> functorial composition of species and cycle index series. The rhs just denotes
> the CIS of H.
>
> Summation over cycles is still not gone.
But over permutations. That's a lot less!
> > Note that taking a permutation to a power can only decrease the length of
> > the
> > longest cycle. (Check this...!)
>
> If you think so...
Yes. Proof: Let \pi be a permutation of [n] and k\in [n]. Let c be the cycle of
\pi that contains k. Applying \pi to k maps k to some other element of c.
> > 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.
So, we would have
cycleTypeSpecies(p: CYCLETYPE, n: Integer): CYCLETYPE == {
cycleType: CYCLETYPE := [];
s := cycleIndexSeries$F;
N := count(n)$F
for k in N..1 by -1 repeat {
divisorsK: List Integer := divisors k;
sum: Integer := 0;
for d in divisorsK repeat {
sum := sum + moebiusMu(k/d)
*coefficient(s, cycleTypePower(p, d));
-- this is very rough... is there such a function coefficient? Or count?
}
cycleType := cons(sum/k, cycleType);
}
cycleType;
}
which is quite ugly, I must confess. (Concerning documentation, I would have
thought that
Thus, we need to compute $(G[\sigma])_k$. Proposition~3 in the same section
reads
\begin{prop}
$$(G[\sigma])_k = \frac{1}{k}\sum_{d\mid k} \mu(k/d) fix G[\sigma^d],$$ where
$\mu$ is the M\"obius function for integers.
\end{prop}
and
Similarly, if $G$ is a species, the longest cylce of $G[\sigma]$ can only be
shorter than the the longest cycle of $\sigma$. (Check this...!)
is enough... (apart from the fact, that the final sentence is wrong)
Martin
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