Definition: Let G(*) be a semigroup. A finite sequence u_1, u_2,... u_n of elements from G is called uniquevalued with value v and length n provided v = u_1 * u_2 *... * u_n and v != u_p(1) * u_p(2) *... * u_p(n) for any nonidentity permutation p of the indices {1, 2,... n}.
In other words, the only way to obtain the unique value v from the elements u_1, u_2,... u_n is by multiplying them in that particular order; any other order always gives a value different from v.
When the length of the sequence is 2, the meaning of "uniquevalued" is equivalent to "the two elements do not commute under *".
I proved that the maximal possible length of a uniquevalued sequence in the monoid M_n(*) of all n X n matrices (with entries in some ring with 1) is exactly 2n1 (that 2n1 is an upper bound follows from the AmitsurLevitzki theorem), providing a positive example that this limit is reached.
I also proved that the maximal possible length of uniquevalued sequences in S_n is 2n3 (using nsimplex and again using the AmitsurLevitzki theorem), but didn't find examples that this limit is really reached. My computer said "yes" for n=2 to 5, but even 6 is too large to compute.
The numbers of uniquevalued sequences in S_n of the maximal length 2n3 form a sequence 1, 2, 12, 576, which seems to coincide with A002860, the number of Latin squares of order n.
