  We present a new family of hook-length formulas for the number of standard
increasing tableaux which arise in the study of factorial Grothendieck
polynomials. In the case of straight shapes our formulas generalize the
classical hook-length formula and Stanley's formula. For skew shapes, our
formulas generalize the Naruse hook-length formula and its $q$-analogues, which
were studied in previous papers of the series.
