# Details of talk

Title | On Lyndon words containing the minimum number of Lyndon factors |
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Presenter | Amy Glen (Murdoch University) |

Author(s) | Amy Glen |

Session | Algebra and Discrete Mathematics |

Time | 14:00:00 2017-09-26 |

Abstract | A \textit{Lyndon word} is a primitive word on an ordered alphabet that is lexicographically less than all of its conjugates. Originally introduced in the context of free Lie algebras (Lyndon 1954, Chen-Fox-Lyndon 1958), Lyndon words have proved to be a useful tool for many problems in combinatorics, such as the construction of \textit{de Bruijn sequences} (Fredricksen-Maiorana 1978) and determining the optimal lower bound for the size of uniform unavoidable sets (Champarnaud \textit{et al.} 2004). A famous theorem concerning Lyndon words asserts that every finite word can be uniquely factorised as a non-increasing product of Lyndon words (Chen-Fox-Lyndon 1958). In this sense, Lyndon words are to combinatorics on words like prime numbers are to number theory. \medskip \hspace*{1em} The minimum number of distinct Lyndon factors that a word of length $n$ can contain is clearly $1$, achieved by $x^n$ where $x$ is a letter. More interesting are the Lyndon words that contain the minimum number of distinct Lyndon factors, a subclass of which consists of the so-called \textit{Fibonacci Lyndon words} (i.e., Lyndon factors of the well-known \textit{infinite Fibonacci word}). Such words will be the focus of this talk. |