[GAP Forum] cycle index and Polya enumeration

[Gordon Royle]

I wish GAP had decent support for polynomials...... This is a bit clunky because I have to use x_1000 for the final variable to sub into the expression... 

cycleIndex computes the cycle index... say 

zg = cycleIndex(g);

Then do

Value(zg, nvars(deg), xvals(deg))

to substitute in the usual "1+x, 1+x^2, 1+x^3 ... " etc...

where "deg" is the degree of the group.


###############################################################################
#
#   Calculates the cycle index of a group directly from the conjugacy classes
#

cycleIndex := function(g) local cl,cs,x,px,term,zg,deg,i;

 zg := 0;
 deg := NrMovedPoints(g);

 cl := ConjugacyClasses(g);

 for x in cl do
  px := Representative(x);
  cs := CycleStructurePerm(px);
  term := Size(x) * Indeterminate(Rationals,1) ^ (deg - NrMovedPoints(px));
#Print(cs,"-");
  for i in [1..deg] do
    if (IsBound(cs[i])) then 
      term := term * Indeterminate(Rationals,i+1)^cs[i]; 
    fi;
  od;
#Print(term,"\n");
  zg := zg + term;
 od;

 return zg/Order(g);

end;




############################################################################
# Returns a list of the first n indeterminates to sub into the cycle index

nvars := function(n) local i, vars;

vars := [];
for i in [1..n] do
  Add(vars,Indeterminate(Rationals,i));
od;

return vars; 

end;


############################################################################
# Returns a list of length n, with entries equal to 1+x, 1+x^2 and so on

xvals := function(n) local i, vals, x;

x := Indeterminate(Rationals,1000);
Print(IndeterminateNumberOfLaurentPolynomial(x),"\n");

vals := [];

for i in [1..n] do
  Add(vals,1+x^i);
od;

return vals;

end;


On 19/12/2010, at 8:22 PM, Hebert Pérez-Rosés wrote:

> Dear all,
> 
> Does anybody have a GAP function to compute the cycle index of a permutation
> group, and perform Polya enumeration?
> 
> Best regards,
> 
> Hebert Perez-Roses
> The University of Newcastle, Australia
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