Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> Huh, the functorial composition of cycle index series in BLL (chapter
> 2.2 formula (12)) looks quite complicated to implement.
>
> Is there perhaps a simpler formula that does not involve "fix", but
> rather only the coefficients of the involved CIS?
Hornegger and Pirastu have a partial (non-working) implementation in
functori.spad. Maybe that helps a bit...
(typpot is not implemented needed for betak, there is no suitable
implementation for create as used in compose, and everything is very
sketchy...)
Very likely you know the following. I just state it here for reference...
To obtain fix F[(G[sigma])_1,(G[sigma])_2,...], you need to compute
(G[sigma])_1, (G[sigma])_2, ...
(these are numbers. In Hornegger Pirastu it is called betak. To compute them,
maybe exercise 2 in Section 2.2 is helpful. I do not understand, but isn't (a)
with m=1 what we need? This looks too simple, though.)
and then extract the coefficient of x1^G[sigma])_1 x2^(G[sigma])_2 ... from
the cycleindexseries of F and multiply with the denominator in Section 1.2,
Equation (19).
Another thing one has to keep in mind is that
fix H[sigma]
only depends on the cycletype of the permutation sigma, not the permutation
itself.
Here is moebiusMu from numtheory.spad:
moebiusMu n ==
n = 1 => 1
t := factor n
for k in factors t repeat
k.exponent > 1 => return 0
odd? numberOfFactors t => -1
1
Martin
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