  Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their
ordinary and double quotients $W / W_I$ and $W_I \backslash W / W_J$, appear in
many contexts in combinatorics and Lie theory, including the geometry and
topology of generalized flag varieties and the symmetry groups of regular
polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the
Coxeter complex of $W$, and is well-studied. In this article we look at a less
studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq
S$. Double coset are not uniquely presented by triples $(I,w,J)$. We describe
what we call the lex-minimal presentation, and prove that there exists a unique
such object for each double coset. Lex-minimal presentations are then used to
enumerate double cosets via a finite automaton depending on the Coxeter graph
for $(W,S)$. As an example, we present a formula for the number of parabolic
double cosets with a fixed minimal element when $W$ is the symmetric group
$S_n$ (in this case, parabolic subgroups are also known as Young subgroups).
Our formula is almost always linear time computable in $n$, and we show how it
can be generalized to any Coxeter group with little additional work. We spell
out formulas for all finite and affine Weyl groups in the case that $w$ is the
identity element.
