Dear Ralf,
sorry, I cannot see the mistake. I don't see any permutations in my code, only
cycle types.
Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> OK, before I go further, just propose the code for what you have
> described. And I guess it will be wrong in any computer language.
But I did write code in my last email: two functions, one computes the cycle
type of the power of a permutation, only given its cycle type:
CYCLETYPE ==> List Integer
cycleTypePower(p: CYCLETYPE, k: Integer): CYCLETYPE == {
cycleType: CYCLETYPE := [];
divisorsK: List Integer := divisors k;
for m in #p..1 by -1 repeat {
sum: Integer := 0;
for d in divisorsK | gcd(m, k/d)=1 repeat {
sum := sum + d * p.(d*m)
}
cylceType := cons(sum, cycleType)
}
cycleType
}
The second one computes the cycle type of F[\sigma], only given the cycle type
of \sigma and the cycleIndexSeries of F.
cycleTypeSpecies(p: CYCLETYPE): CYCLETYPE == {
cycleType: CYCLETYPE := [];
s := cycleIndexSeries;
n := lcm([i for i in 1..#p | p.i ~= 0]);
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;
}
To get the cycleIndexSeries of F \square G, you have to compute
\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*}
where
\fix(F\square G)[n_1, n_2, \dots]
is
coefficient(cycleIndexSeries$F, cycleTypeSpecies [n1, n2,...])
(I'm not sure what coefficient(cycleIndexSeries$F, ...) does exactly. If you
include the denominator 1^n1 n1! 2^n2 n2! ... in the coefficient you have to
multiply by it again... The same remark holds for the computation of
cycleTypeSpecies)
Does this answer your question? Did I miss something?
Martin
-------------------------------------------------------------------------
Take Surveys. Earn Cash. Influence the Future of IT
Join SourceForge.net's Techsay panel and you'll get the chance to share your
opinions on IT & business topics through brief surveys-and earn cash
http://www.techsay.com/default.php?page=join.php&p=sourceforge&CID=DEVDEV
_______________________________________________
Aldor-combinat-devel mailing list
Aldor-combinat-devel@lists.sourceforge.net
https://lists.sourceforge.net/lists/listinfo/aldor-combinat-devel
