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 ( (log _2t )^2 )=6t^4 (log _2 (t + 1 ) )^2/π (t^2-1 ) (2t-1/2log _2 (π t ) )^2More