  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
$\sigma$, 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
$x^n$ 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
$\lceil\log_{\phi}(n) + 1\rceil$, where $\phi$ denotes the golden ratio $(1 +
\sqrt{5})/2$. 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
$n$, $n\ne 6$, 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.
