The Number of Hierarchical Orderings
<|reference_start|>The Number of Hierarchical Orderings: 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 m <= n 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.<|reference_end|>
@article{sloane2003the,
    title={The Number of Hierarchical Orderings},
    author={N. J. A. Sloane, Thomas Wieder},
    journal={Order, 21 (2004), 83-89},
    year={2003},
    archivePrefix={arXiv},
    eprint={math/0307064},
    primaryClass={math.CO cs.IT math.IT}
}