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

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 \(t=\omega \left (\frac {n^{1/8}}{\left (\log _{2}n\right )^{3/4}}\right )\).

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Acknowledgements

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

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Correspondence to Shen-Fu Tsai.

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Tsai, S. A Simple Upper Bound on the Number of Antichains in [t]n. Order 36, 507–510 (2019). https://doi.org/10.1007/s11083-018-9480-5

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Keywords

  • Partially ordered set
  • Dedekind’s problem
  • Monotonic Boolean function