  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.
