Properties of plactic monoid centralizers

Let u be a word over the positive integers P. Motivated by a question involving crystal graphs, Sagan and Wilson initiated the study of the centralizer of u in the plactic monoid which is the set C(u) = {w | uw is Knuth equivalent to wu}. In particular, they conjectured the following stability phenomenon: for any u there is a positive integer K depending only on u such that C(u^k) = C(u^K) for k >= K. We prove that this property holds for various u including words consisting of only ones and twos, as well as permutations. Sagan and Wilson also considered c_{n,m}(u) which is the number of w in C(u) of length n and maximum at most m. They showed that c_{n,m}(1) is a polynomial in m of degree n-1 and conjectured properties of the coefficients when it is expanded in a binomial coefficient basis. We prove some of these conjectures, for example, that the coefficients are always nonnegative integers.