2025/12/24 by Bruce E. Sagan, Chenchen Zhao, Sagan, Bruce E. +1
Computer Science · Mathematics · #05A15 (Secondary) #05E99 (Primary) 05A05 #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #semigroups and automata theory
paper · doi:10.48550/arxiv.2512.21401
openalex publication_date 2025/12/24 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28
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(uk) = C(uK) 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 cn,m(u) which is the number of w in C(u) of length n and maximum at most m. They showed that cn,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.