I have a routine which returns a certain result. In the simplest case the result should be p[[1]].

The result is indeed p[[1]]. However, when I ask whether the result is equal to p[[1]], Maple says false.

Why does is not recognize these two things as being equal?

This was very surprising!!

x:=3;
                               3
g:=x;h:=x^2;
                               3
                               9
subs(x=4,g);subs(x=4,h);
                               4
                               9

This is inconsistent! After g and h have been assigned values, they do not depend on x anymore, so g should not be turned into 4 by subs

I want to solve for the coefficients in some multivariate polynomials by equating them to other known multivariate polynomials.

Something like this. I have

p[1]=(a+b)*x^2+(a+c)*x*y+d*z;

p[2]=(a-b)*x+e*y*z+f*z^2;

I want to impose that p[1]=x^2+2*x*y+3*z and that p[2]=x+4*y*z and I want Maple to tell me the values of (a,b,c,d,e,f).

Sounds simple enough, but I have not been able to do it

Let G be a permutation group like S3. I have a loop in which x runs over the elements of this group.

When I ask the cycle type of x or the number of orbits of x, Maple gives an incorrect result in some cases. This is because when x=(12), for example, Maple considers x to be an element of the permutation group S2 instead of S3, i.e. it fails to understand that (12) should really be (12)(3) in this case.

Is there a way I can instruct Maple to consider the permutation (12) as a member of S3 instead of S2? 

I am posting the code as requested. Something like this:

with(GroupTheory);

for x in Elements(S3) do print(x, numelems(Orbits(PermutationGroup(x)))) end do;

(I don't know how to format this, sorry)

This code is simply asking how many orbits exist in the action of each element of S3. The result will be 1 for x=(12), however, because Maple will assume that (12) is acting on the set {1,2}, even though I am considering (12) as an element of S3. I want it to consider (12)=(12)(3) and let it act on the set {1,2,3}, thereby producing 2 orbits, not 1.

I want to sum over several variables at once.

For example, something like

add(add(add(f(k1,k2,k3),k1=range1),k2=range2),k3=range3);

sums over 3 variables. But this can only be done if I know the number of variables in advance. I want to write a code that will sum over m variables, with m being supplied on demand.

Maple can do this for integrals. If I write

int(f(seq(k[i],i=1..m)),[seq(k[i]=range[i],i=1..m)]);

the integral is computed over m variables, for any m.

How can something like this be implemented for summation?