2017/01/04 by Amy Glen, Glen, Amy, Jamie Simpson +3
Biochemistry, Genetics and Molecular Biology · Computer Science · #68R15 #Algorithms and Data Compression #Combinatorics (math.CO) #DNA and Biological Computing #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1701.00928
openalex publication_date 2017/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we determine the maximum number of distinct Lyndon factors that a word of length n can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length n on an alphabet of size σ, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length n is 1 and the minimum total number is n, with both bounds being achieved by xn where x is a letter. A more interesting question to ask is what is the minimum number of distinct Lyndon factors in a Lyndon word of length n? In this direction, it is known (Saari, 2014) that an optimal lower bound for the number of distinct Lyndon factors in a Lyndon word of length n is \lceillogϕ(n) + 1\rceil, where ϕ denotes the golden ratio (1 + √(5))/2. Moreover, this lower bound is attained by the so-called finite "Fibonacci Lyndon words", which are precisely the Lyndon factors of the well-known "infinite Fibonacci word" -- a special example of a "infinite Sturmian word". Saari (2014) conjectured that if w is Lyndon word of length n, n≠ 6, containing the least number of distinct Lyndon factors over all Lyndon words of the same length, then w is a Christoffel word (i.e., a Lyndon factor of an infinite Sturmian word). We give a counterexample to this conjecture. Furthermore, we generalise Saari's result on the number of distinct Lyndon factors of a Fibonacci Lyndon word by determining the number of distinct Lyndon factors of a given Christoffel word. We end with two open problems.