Let u be a word over the positive integers \({\mathbb {P}}\) . 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 \( C(u) = \{w \mid uw \text { is Knuth equivalent to } wu\}. \) 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 \(k\ge K\) . 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.
