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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.
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    Your title does not reflect your question.2011-03-25
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    If two words are not cyclically related, does that mean their traces are different or could be different?2011-03-25
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    Note that your examples don't match the original restrictions (they have only two $A$s, two $B$s, and one $C$).2011-03-25
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    If we consider only the equivalence of strings under cyclic permutations, counting the distinct equivalence classes is a type of [necklace problem](http://en.wikipedia.org/wiki/Necklace_(combinatorics)). If certain matrices commute, then this would create additional equivalences, complicating the counting.2011-03-25
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    @Arturo Thanks for pointing out the typo with the number of Bs..corrected that.2011-03-25

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