We give an explicit natural identification between the quiver coefficients of
Buch and Fulton, decomposition coefficients for Schubert polynomials, and the
Schubert structure constants for flag manifolds.
This is also achieved in K-theory where we give a direct argument that
the decomposition coefficients have alternating signs, based on a theorem of
Brion, which then implies that the quiver coefficients have alternating
signs.
Our identification shows that known combinatorial formulas for the latter two
numbers give formulas for the quiver coefficients.