2023
DOI: 10.48550/arxiv.2301.13829
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The Deepest Cycle of a Random Mapping: a Problem Proposed by Steven Finch

Abstract: The corresponding graph of T is a union of disjoint connected unicyclic components. We assume that each T ∈ Tn is chosen uniformly at random (i.e., with probability n −n ). The deepest cycle of T is contained within its largest component. Let νn = νn(T ) denote the length of the deepest cycle in T ∈ Tn. In this paper, we find the limits of the expectation and variance of νn/ √ n as n → ∞. For n large enough, we also show that nearly 55% of all cyclic vertices of a random mapping T ∈ Tn lie in its deepest cycle… Show more

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