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