  A $1$-prefix normal word is a binary word with the property that no factor
has more $1$s than the prefix of the same length; a $0$-prefix normal word is
defined analogously. These words arise in the context of indexed binary jumbled
pattern matching, where the aim is to decide whether a word has a factor with a
given number of $1$s and $0$s (a given Parikh vector). Each binary word has an
associated set of Parikh vectors of the factors of the word. Using prefix
normal words, we provide a characterization of the equivalence class of binary
words having the same set of Parikh vectors of their factors.
  We prove that the language of prefix normal words is not context-free and is
strictly contained in the language of pre-necklaces, which are prefixes of
powers of Lyndon words. We give enumeration results on $\textit{pnw}(n)$, the
number of prefix normal words of length $n$, showing that, for sufficiently
large $n$, \[ 2^{n-4 \sqrt{n \lg n}} \le \textit{pnw}(n) \le 2^{n - \lg n + 1}.
\]
  For fixed density (number of $1$s), we show that the ordinary generating
function of the number of prefix normal words of length $n$ and density $d$ is
a rational function. Finally, we give experimental results on
$\textit{pnw}(n)$, discuss further properties, and state open problems.
