Flagged Grothendieck enumerators and ribbon decomposition determinants

We establish Hamel--Goulden-type ribbon decomposition determinantal formulas for a new class of symmetric functions called flagged Grothendieck enumerators. In particular, these functions generalize both canonical refined stable Grothendieck polynomials and their duals. The proof uses Talaska's formula via flows on planar lattice graphs with systems of paths and cycles.