Pairs of majority-decomposing functions

We are interested in decompositions 〈xnf1f2〉 of the majority function over n odd arguments x1,…,xn such that f1 and f2 do not depend on xn. In this paper, we derive the conditions for f1 and f2 that satisfy the decomposition. Such decompositions play a central role in finding optimum majority-3 networks for the majority-n function.