2024/10/27 by Bruce E. Sagan, Sagan, Bruce E., Wilson, Alexander N. · 1 citation
Biochemistry, Genetics and Molecular Biology · #Plant Reproductive Biology
paper · pdf · doi:10.48550/arxiv.2410.20460
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 C(u) = w | uw is Knuth-equivalent to wu. 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 cn,m(u), the number of w in C(u) of length n with max w at most 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, cn,m(u) is a polynomial in m with certain degree and leading coefficient. We end with various conjectures and directions for further research.