  We give an explicit Grobner basis for the ideal of the tangent cone at any
T-fixed point of a Richardson variety in the Grassmannian, thus generalizing a
result of Kodiyalam-Raghavan and Kreiman-Lakshmibai. Our proof is based on a
generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we
call the bounded RSK (BRSK). We use the Grobner basis result to deduce a
formula which computes the multiplicity of the Richardson variety at any
T-fixed point by counting families of nonintersecting lattice paths, thus
generalizing a result first proved by Krattehthaler.
