{
  "id": "2206.05613",
  "version": "v1",
  "published": "2022-06-11T20:51:10.000Z",
  "updated": "2022-06-11T20:51:10.000Z",
  "title": "Barcode Posets: Combinatorial Properties and Connections",
  "authors": [
    "Edgar Jaramillo-Rodriguez"
  ],
  "comment": "20 pages, 5 figures",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-11T20:51:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "combinatorial properties",
      "multinomial newman lattice",
      "barcode posets",
      "connections",
      "persistent homology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}