${\mathbb{N}}^d$-Indexed persistence modules, higher dimensional partitions and rank invariants

We study decomposable ${\mathbb{N}}^d$-indexed persistence modules via higher dimensional partitions. Their barcodes are defined in terms of the extended interior of the corresponding Young diagrams. For two decomposable ${\mathbb{N}}^d$-indexed persistence modules, we present a necessary and sufficient condition, in terms of the partitions, for their rank invariants to be the same. This generalizes the well-known fact that for an $\mathbb{N}$-indexed persistence module, its barcode and its rank invariant determine each other, i.e., the rank invariant is a complete invariant.