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 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.
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References
Andrews, G. E.: The Theory of Partitions, Addison-Wesley, Reading, MA, 1976.
Bergson, F., Labelle, G. and Leroux, P.: Combinatorical Species and Tree-like Structures, Encyclopedia of Mathematics and its Applications 67, Cambridge University Press, Cambridge, 1998.
Cameron, P. J.: Some sequences of integers, Discrete Math. 75 (1989), 89-102.
Graham, R. L., Knuth, D. E. and Patashnik, O.: Concrete Mathematics, 2nd edn, Addison-Wesley, Reading, MA, 1994.
INRIA Algorithms Project: Algolib: The Algorithms Project’s Library, published electronically at pauillac.inria.fr/algo/libraries/, 2003.
Motzkin, T. S.: Sorting numbers for cylinders and other classification numbers, in T. S. Motzkin (ed.), Combinatorics, Proc. Symp. Pure Math. 19, Amer. Math. Soc., Providence, RI, 1971, pp. 167-176.
Odlyzko, A. M.: Asymptotic enumeration methods, in R. L. Graham, M. Grötschel and L. Lovász (eds.), Handbook of Combinatorics, Vol. 2, Elsevier, Amsterdam, and MIT Press, Cambridge, MA, 1995, pp. 1063-1229.
Riordan, J.: An Introduction to Combinatorial Analysis, Wiley, New York, 1958.
Salvy, B.: Examples of automatic asymptotic expansions, Report 114, Institut National de Recherche en Informatique et en Automatique, 1989.
Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences, published electronically at www.research.att.com/~njas/sequences/, 2003.
Wilf, H. S.: Generatingfunctionology, 2nd edn, Academic Press, San Diego, 1994.
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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