Let u be a word over the positive integers. Motivated in part by a question from representation theory, we study the centralizer set of u which is \(\begin{aligned} C(u) = \{w \mid uw\text { is Knuth-equivalent to }wu\}. \end{aligned}\) In particular, we give various necessary conditions for w to be in C(u). We also characterize C(u) when u has few letters, when it has a single repeated entry, or when it is a certain type of decreasing sequence. We consider \(c_{n,m}(u)\) , the number of \(w\in C(u)\) of length n with \(\max w\le m\) . We prove that for \(|u|=1\) the value of this function depends only on the relative sizes of u and m and not on their actual values. And for various u we use Stanley’s theory of poset partitions to show that, for fixed n, \(c_{n,m}(u)\) is a polynomial in m with certain degree and leading coefficient. We end with various conjectures and directions for further research.
