If I have say $2$ A matrices, $2$ B matrices and $1$ C matrix then I would like to know how many distinct traces of products of all of them can be formed.
Assume that $A$, $B$ and $C$ don't commute with each other.
Like $AABBC$, $CAABB$ and $BCAAB$ are distinct products which will have the same trace but $ABABC$ has a different trace.
I would like to know how to count the number of products upto having the same trace.
- Though I don't need it at this point I would be curious to know how the problem might get complicated if one assumes any or more of the pair of matrices commute.