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The Number of Hierarchical Orderings

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Abstract

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into mn subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case.

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Sloane, N.J.A., Wieder, T. The Number of Hierarchical Orderings. Order 21, 83–89 (2004). https://doi.org/10.1007/s11083-004-9460-9

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  • DOI: https://doi.org/10.1007/s11083-004-9460-9

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