Counting Lyndon factors
Amy Glen, Jamie Simpson, W. F. Smyth

TL;DR
This paper investigates the maximum and expected counts of Lyndon factors in words, explores bounds for Lyndon factors in Lyndon words, and challenges existing conjectures with new counterexamples and generalizations.
Contribution
It provides formulas for Lyndon factors, refutes a conjecture about Christoffel words, and extends results to Christoffel words from Fibonacci Lyndon words.
Findings
Maximum number of Lyndon factors in words of length n determined
Formulas for expected total and distinct Lyndon factors derived
Counterexample provided to Saari's conjecture on Christoffel words
Abstract
In this paper, we determine the maximum number of distinct Lyndon factors that a word of length can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length on an alphabet of size , 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 is and the minimum total number is , with both bounds being achieved by where 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 ? 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 is , where denotes the golden ratio . Moreover, this lower bound is attained…
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Taxonomy
Topicssemigroups and automata theory · Algorithms and Data Compression · DNA and Biological Computing
