Properties of plactic monoid centralizers

Let u be a word over the positive integers P 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) = uw is Knuth equivalent to wu\. C ( u ) = { w ∣ u w is Knuth equivalent to w u } . In particular, they conjectured the following stability phenomenon: for any u there is a positive integer K depending only on u such that C(uᵏ) = C(uK) C ( u k ) = C ( u K ) for k≥ K 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 cn,m(u) c n , m ( u ) which is the number of w∈ C(u) w ∈ C ( u ) of length n and maximum at most m . They showed that cn,m(1) c n , m ( 1 ) is a polynomial in m of degree $$n-1$$ 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.