Barcode Posets: Combinatorial Properties and Connections

Edgar Jaramillo-Rodriguez

Published 2022-06-11Version 1

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 (TDA), where they serve as summaries of the persistent homology groups of a filtration. The combinatorial properties of barcodes have been studied in the context of interval orders and interval graphs. Recently, Kanari, Garin, and Hess discovered a natural mapping between the space of barcodes with $n+1$ bars and the symmetric group $\mathfrak{S}_n$, relating the combinatorial structure of barcodes to TDA. In this paper, we define a new family of maps from the space of barcodes with $n$ bars to the permutations of various multisets. These sets of permutations are known as the multinomial Newman lattices. Each map produces a new combinatorial invariant for a given barcode and we show that these invariants have a poset structure derived from the multinomial Newman lattice. We show that these posets are in fact order-isomorphic to principal ideals of their multinomial Newman lattice, and consequently they are graded face-lattices of polytopes. We call these posets the barcode lattices and show that for a large class of barcodes, these invariants can provide bounds on the Wasserstein and bottleneck distances between a pair of barcodes.