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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymptotic enumeration of partial orders on a finite set


Authors: D. J. Kleitman and B. L. Rothschild
Journal: Trans. Amer. Math. Soc. 205 (1975), 205-220
MSC: Primary 05A15; Secondary 06A10
DOI: https://doi.org/10.1090/S0002-9947-1975-0369090-9
MathSciNet review: 0369090
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Abstract: By considering special cases, the number $ {P_n}$ of partially ordered sets on a set of $ n$ elements is shown to be $ (1 + O(1/n)){Q_n}$, where $ {Q_n}$ is the number of partially ordered sets in one of the special classes. The number $ {Q_n}$ can be estimated, and we ultimately obtain

$\displaystyle {P_n} = \left( {1 + O\left( {\frac{1}{n}} \right)} \right)\left( ... ...{{2^i} - 1} \right)}^j}{{\left( {{2^j} - 1} \right)}^{n - i - j}}} } } \right).$


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DOI: https://doi.org/10.1090/S0002-9947-1975-0369090-9
Article copyright: © Copyright 1975 American Mathematical Society