  We introduce a q-analogue of the classical Zeta polynomial of finite
partially ordered sets, as a polynomial in one variable x with coefficients
depending on the indeterminate q. We prove some properties of this polynomial
invariant, including its behaviour with respect to duality, product and
disjoint union. The leading term is a q-analogue of the number of maximal
chains, but not always with non-negative coefficients. The value at q=0 turns
out to be essentially the characteristic polynomial.
