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, Volume 36, Issue 3, pp 507–510 | Cite as

A Simple Upper Bound on the Number of Antichains in [t]n

  • Shen-Fu TsaiEmail author
Article
  • 36 Downloads

Abstract

In this paper for t > 2 and n > 2, we give a simple upper bound on a ([t]n), the number of antichains in chain product poset [t]n. When t = 2, the problem reduces to classical Dedekind’s problem posed in 1897 and studied extensively afterwards. However few upper bounds have been proposed for t > 2 and n > 2. The new bound is derived with straightforward extension of bracketing decomposition used by Hansel for bound \(3^{n\choose \lfloor n/2\rfloor }\) for classical Dedekind’s problem. To our best knowledge, our new bound is the best when \({\Theta }\left (\left (\log _{2}t\right )^{2}\right )=\frac {6t^{4}\left (\log _{2}\left (t + 1\right )\right )^{2}}{\pi \left (t^{2}-1\right )\left (2t-\frac {1}{2}\log _{2}\left (\pi t\right )\right )^{2}}<n\) and \(t=\omega \left (\frac {n^{1/8}}{\left (\log _{2}n\right )^{3/4}}\right )\).

Keywords

Partially ordered set Dedekind’s problem Monotonic Boolean function 

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Notes

Acknowledgements

The author would like to thank the anonymous reviewers for their valuable comments and suggestions.

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Google Inc.KirklandUSA

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