In this paper, we construct formulae for the k-th elementary symmetry polynomial of n Boolean variables, using only conjunction and disjunction, which for fixed k are of size O(n log n), with the construction taking time polynomial in n. We also prove theorems involving n log n/spl middot/(polynomial in k) upper bounds on such formulae. Our methods involve solving the following combinatorial problem: for fixed k and any n construct a collection of r=O(log n) functions f/sub 1/,...,f/sub r/ from {1,...,n} to {1,...,K} such that any subset of {1,...,n} of order k is mapped 1-1 to {1,...,k} by at least one f/sub i/..