  We give four positive formulae for the (equioriented type A) quiver
polynomials of Buch and Fulton. All four formulae are combinatorial, in the
sense that they are expressed in terms of combinatorial objects of certain
types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe
dreams (also known as rc-graphs). Three of our formulae are multiplicity-free
and geometric, meaning that their summands have coefficient 1, and correspond
bijectively to components of a torus-invariant scheme. The remaining (presently
non-geometric) formula was conjectured for by Buch and Fulton in terms of
factor sequences of Young tableaux; our proof of it proceeds by way of a new
characterization of the tableaux counted by quiver constants. All four formulae
come naturally in ``doubled'' versions, two for `double quiver polynomials',
and the other two for their stable versions, the `double quiver functions',
where setting half the variables equal to the other half specializes to the
ordinary case.
