Combinatorial methods for barcode analysis

A barcode is a finite multiset of closed intervals on the real line. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. We introduce a new combinatorial invariant associated to barcodes by mapping each barcode to a multipermutation, i.e., a permutation of some multiset, which captures the overlapping arrangement of its bars. We call the set all such multipermutations the space of combinatorial barcodes. We define an order on this space and show that the resulting poset is a graded lattice. The cover relations in this lattice can also be used to determine the set of barcode bases of persistence modules. We explore some connections between combinatorial barcodes, trapezoidal words, and Stirling permutations. Finally, we generalize this construction, producing an entire family of multipermutation invariants of barcodes. For a large class of barcodes, these multipermutations provide bounds on the Wasserstein and bottleneck distances between pairs of barcodes, thereby linking combinatorial barcodes to continuous metrics on barcodes. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023.