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

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⌊n/2⌋) for classical Dedekind’s problem. To our best knowledge, our new bound is the best when Θ((log2t)2)=6t4(log2(t+1))2π(t2−1)(2t−12log2(πt))2<n and t=ω(n1/8(log2n)3/4).