  Buch and Fulton conjectured the nonnegativity of the quiver coefficients
appearing in their formula for a quiver variety. Knutson, Miller and Shimozono
proved this conjecture as an immediate consequence of their ``component
formula''. We present an alternative proof of the component formula by
substituting combinatorics for Grobner degeneration. We relate the component
formula to the work of Buch, Kresch, Tamvakis and the author where a
``splitting'' formula for Schubert polynomials in terms of quiver coefficients
was obtained. We prove analogues of this latter result for the type
BCD-Schubert polynomials of Billey and Haiman.
