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We study the decomposition of zero-dimensional persistence modules, viewed asfunctors valued in the category of vector spaces factorizing through sets.Instead of working directly at the level of vector spaces, we take a step backand first study the decomposition problem at the level of sets.  This approach allows us to define the combinatorial notion of rooted subsets.In the case of a filtered metric space $M$, rooted subsets relate theclustering behavior of the points of $M$ with the decomposition of theassociated persistence module. In particular, we can identify intervals in sucha decomposition quickly. In addition, rooted subsets can be understood as ageneralization of the elder rule, and are also related to the notion ofconstant conqueror of Cai, Kim, M\'emoli and Wang. As an application, we give alower bound on the number of intervals that we can expect in the decompositionof zero-dimensional persistence modules of a density-Rips filtration inEuclidean space: in the limit, and under very general circumstances, we canexpect that at least 25% of the indecomposable summands are interval modules.

Decomposition of zero-dimensional persistence modules via rooted subsets