Counting Lyndon Factors

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 a lower bound for the number of distinct Lyndon factors in a Lyndon word of length n is dlogφ(n)+1e, where φ denotes the golden ratio (1+ √ 5)/2. Moreover, this lower bound is sharp when n is a Fibonacci number and is attained by the so-called finite Fibonacci Lyndon words, which are precisely the Lyndon factors of the wellknown infinite Fibonacci word f (a special example of an infinite Sturmian word). Saari (2014) conjectured that if w is Lyndon word of length n, n 6= 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. the electronic journal of combinatorics 24(3) (2017), #P3.28 1