On 03/13/2007 09:48 AM, Martin Rubey wrote:
> Sorry for following up on my own mail...
>
> Martin Rubey <[EMAIL PROTECTED]> writes:
>
>> -- Idea 2
>> ---------------------------------------------------------------------
>>
>>
>> Similar to the current approach in AC of computing cycle index
>> series via, it might be possible to compute only certain
>> coefficients of the cycle index series.
>
> So, I tried the easiest non-trivial example, namely that of Times:
>
> We'd have
>
> [x^\sigma] Z_{f*g} = \sum_{\tau <= \sigma} [x^\tau] Z_f
> [x^{\sigma-\tau}] Z_g
>
> where
>
> * \sigma is a cycle type
>
> * x^\sigma = x1^{\sigma_1} x2^{\sigma_2} ...
>
> * [x^\sigma] Z_f denotes the coefficient in Z_f
>
> * \tau <= \sigma is to be understood componentwise.
>
> I believe that this would imply a significant speedup in
> functorialCompose.
>
> Ralf, don't you do something like this internally anyway, in order to
> obtain the product of cycle index series?
It depends on how you see it. You probably know how you multiply
univariate power series. Now think of the coefficient as a (weighted)
homogeneous polynomial in the variables x1,...,xn. Multiplication of two
CIS is done exactly that way.
If I do as you suggest, I would have to recompute [x^\sigma] Z_{f*g}
each time I need it. Or perhaps you tell me how you then compute the
n-th coefficient of the isomorphismType series from the CIS.
> (Well, it doesn't seem so. Maybe that's a point for having
> multivariate series...) In this case, it would only be a
> matter of refactoring code.
What has that to do with multivariate series. Actually, a CIS is already
a multivariate series. ;-)
Ralf
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