Planarity in ROMDDs of multiple-valued symmetric functions

We show that a multiple-valued symmetric function has a planar ROMDD (reduced ordered multiple-valued decision diagram) if and only if it is a pseudo-voting function. We show that the number of such functions is (r-1)(n+r, n+1) where r is the number of logic values and n is the number of variables. It follows from this that the fraction of symmetric multiple-valued functions that have planar ROMDDs approaches 0 as n approaches infinity. Further, we show that the worst case and average number of nodes in planar ROMDDs of symmetric functions is n²(1/2-1/2r) and n²(1/2-1/(r+1)), respectively, when n is large