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.
Citation: Mehdi Nategh, Zhenbo Qin, Shuguang Wang. $ \mathbb N^d $-Indexed persistence modules, higher dimensional partitions and rank invariants[J]. AIMS Mathematics, 2025, 10(11): 25589-25605. doi: 10.3934/math.20251133
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.
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Mehdi Nategh, Zhenbo Qin, Shuguang Wang. $ \mathbb N^d $-Indexed persistence modules, higher dimensional partitions and rank invariants[J]. AIMS Mathematics, 2025, 10(11): 25589-25605. doi: 10.3934/math.20251133
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