On the number of bipolar Boolean functions

A Boolean function is bipolar iff it is monotone or anti-monotone 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 closed-form 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.