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We introduce edge labeled Young tableaux. Our main results provide acorresponding analogue of [Sch\"{u}tzenberger '77]'s theory of jeu de taquin.These are applied to the equivariant Schubert calculus of Grassmannians.Reinterpreting, we present new (semi)standard tableaux to study factorial Schurpolynomials, after [Biedenharn-Louck '89], [Macdonald '92] and [Goulden-Greene'94] and others.  Consequently, we obtain new combinatorial rules for the Schubert structurecoefficients, complementing work of [Molev-Sagan '99], [Knutson-Tao '03],[Molev '08] and [Kreiman '09]. We also describe a conjectural generalization ofone of our rules to the equivariant K-theory of Grassmannians, extending workof [Thomas-Yong '07]. This conjecture concretely realizes the "positivity"known to exist by [Anderson-Griffeth-Miller '08]. It provides an alternative tothe conjectural rule of Knutson-Vakil reported in [Coskun-Vakil '06].

Equivariant Schubert calculus and jeu de taquin