2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67945We prove K-theoretic generalizations of the component formulas of Knutson, Miller, and Shimozono, and deduce that K-theoretic quiver coefficients have alternating signs. We also prove new variants of the factor sequences conjecture, and a conjecture of Knutson, Miller, and Shimozono stating that the double ratio formula agrees with the original quiver formulas. For completeness we include a short proof of the ratio formula for quiver varieties.This paper has been reorganized to accommodate new variations of the factor sequences conjecture. A short proof of the ratio formula has also been added for completeness. 20 pagesCombinatoricsAlgebraic Geometry05E15 (Primary) 14M15, 14M12, 19E08 (Secondary)text