  We use tools from geometric group theory to produce a stratification of the
space $\mathcal{B}_n$ of barcodes with $n$ bars. The top-dimensional strata are
indexed by permutations associated to barcodes as defined by Kanari, Garin and
Hess. More generally, the strata correspond to marked double cosets of
parabolic subgroups of the symmetric group $Sym_n$. This subdivides
$\mathcal{B}_n$ into regions that consist of barcodes with the same averages
and standard deviations of birth and death times and the same permutation type.
We obtain coordinates that form a new invariant of barcodes, extending the one
of Kanari-Garin-Hess. This description also gives rise to metrics on
$\mathcal{B}_n$ that coincide with modified versions of the bottleneck and
Wasserstein metrics.
