  A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i,
d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis,
where they serve as summaries of the persistent homology groups of a
filtration. The combinatorial properties of barcodes have also been studied,
mainly in the context of interval orders and interval graphs. In this paper, we
define a new family of maps from the space of barcodes with $n$ bars to the
permutation sets of various multisets, known as multipermutations. These
multipermutations provide new combinatorial invariants on the space of
barcodes. We then define an order relation on these multipermutations, which we
show can be interpreted as a crossing number for barcodes, reminiscent of
T\'{u}ran's crossing number for graphs. Next, we show that the resulting posets
are order-isomorphic to principal ideals of a well known poset known as the
multinomial Newman lattice. Consequently, these posets form the graded
face-lattices of polytopes, which we refer to as barcode lattices or barcode
polytopes. Finally, we show that for a large class of barcodes, these
invariants can provide bounds on the Wasserstein and bottleneck distances
between a pair of barcodes, linking these discrete invariants to continuous
metrics on barcodes.
