A Combinatorial Formula for the Bigraded Betti Numbers

Abstract

It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In contrast, Carlsson and Zomorodian showed that $n$-parameter persistence modules have no such "nice" classification when $n>1$; every discrete invariant is incomplete. Despite their incompleteness, discrete invariants can still provide insight into the properties of multiparameter persistence modules. A well-studied discrete invariant for $2$-parameter persistence modules is the bigraded Betti numbers. Through commutative algebra techniques, it is known that the bigraded Betti numbers of a $2$-parameter persistence module $M$ can be recovered from the barcodes of certain zigzag modules within $M$ via a simple combinatorial formula. We present an alternate proof of this formula that relies only on basic linear algebra.

12 pages. Updated to better clarify what was previously known and what aspects are our contributions