On the similarity classes among products of m nonsingular matrices in various orders

It is shown that for any k∈{1,2,…,(m−1)!} there exist m invertible complex matrices such that among the m! products Aσ=Aσ(1)Aσ(2)⋯Aσ(m), σ∈Sm, exactly k different similarity classes occur. The cases in which the matrices Ai are upper triangular or are 2-by-2 are considered in detail. In the former case, it is shown that any m−1 unispectral matrices with common eigenvalue may be found among the Aσ, and in the latter case it is shown explicitly how to achieve (m−1)! and (m−1)!2 similarity classes, as well as any number from 1 to 6 when m=4. Other particular results are given, as well as a discussion of further natural questions.