On the number of bipolar Boolean functions

A Boolean function is bipolar iff it is monotone or antimonotone in each of its arguments. We investigate the number b(n) of n-ary bipolar Boolean functions. We present an (almost) closed-form expression for b(n) that uses the number a(n) of antichain covers of an n-element set. This is closely related to Dedekind’s problem, which can be rephrased as determining the number d(n) of Boolean functions that are monotone in all arguments. Indeed, a closedform solution of a(n) would directly yield a closed-form solution of d(n), suggesting that determining a(n) is a non-trivial problem of itself.