  We present an elementary method for proving enumeration formulas which are
polynomials in certain parameters if others are fixed and factorize into
distinct linear factors over Z. Roughly speaking the idea is to prove such
formulas by ``explaining'' their zeros using an appropriate combinatorial
extension of the objects under consideration to negative integer parameters. We
apply this method to prove a new refinement of the Bender-Knuth
(ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself.
This is probably the most elementary way to prove this result currently known.
Furthermore we adapt our method to q-polynomials, which allows us to derive
generating function results as well. Finally we use this method to give another
proof for the enumeration of semistandard tableaux of a fixed shape, which is
opposed to the Bender-Knuth (ex-)Conjecture refinement a multivariate
application of our method.
