>> Summation over cycles is still not gone.
>
> But over permutations. That's a lot less!
Arrrggghhh. Of course my sentence should have read:
"Summation over permutations is still not gone."
> 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)
Oh, I have not guessed that and the function name "cycleTypeSpecies"
tells me nothing at all. Not even from the documentation you give I
understand the input/output specification.
Ralf
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