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We prove a restriction of an analogue of the Robinson--Schensted--Knuthcorrespondence for semi-skyline augmented fillings, due to Mason, to multisetsof cells of a staircase possibly truncated by a smaller staircase at the upperleft end corner, or at the bottom right end corner.  The restriction to be imposed on the pairs of semi-skyline augmented fillingsis that the pair of shapes, rearrangements of each other, satisfies aninequality in the Bruhat order, w.r.t. the symmetric group, where one shape isbounded by the reverse of the other. For semi-standard Young tableaux theinequality means that the pair of their right keys is such that one key isbounded by the Sch\"utzenberger evacuation of the other. This bijection is thenused to obtain %recover an expansion formula of the non-symmetric Cauchykernel, over staircases or truncated staircases, in the basis of Demazurecharacters of type $A$, and the basis of Demazure atoms. The expansion impliesLascoux expansion formula, when specialised to staircases or truncatedstaircases, and make explicit, in the latter, the Young tableaux in theDemazure crystal by interpreting Demazure operators via elementary bubblesorting operators acting on weak compositions.

An analogue of the Robinson-Schensted-Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases