Nonnegative Hall Polynomials

The number of subgroups of type n and cotype v in a finite abelian p-group of type A is a polynomial g$v(p) with integral coefficients. We prove g u v ( p ) has nonnegative coefficients for all partitions p and v if and only if no two parts of A differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections f that associate to each subgroup a vector dominated componentwise by A. The nonzero components of f(H) are the parts of /, the type of H; if no two parts of A differ by more than one, the nonzero components of A f(H] are the parts of v, the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.