  We prove a restriction of an analogue of the Robinson--Schensted--Knuth
correspondence for semi-skyline augmented fillings, due to Mason, to multisets
of cells of a staircase possibly truncated by a smaller staircase at the upper
left end corner, or at the bottom right end corner.
  The restriction to be imposed on the pairs of semi-skyline augmented fillings
is that the pair of shapes, rearrangements of each other, satisfies an
inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is
bounded by the reverse of the other. For semi-standard Young tableaux the
inequality means that the pair of their right keys is such that one key is
bounded by the Sch\"utzenberger evacuation of the other. This bijection is then
used to obtain %recover an expansion formula of the non-symmetric Cauchy
kernel, over staircases or truncated staircases, in the basis of Demazure
characters of type $A$, and the basis of Demazure atoms. The expansion implies
Lascoux expansion formula, when specialised to staircases or truncated
staircases, and make explicit, in the latter, the Young tableaux in the
Demazure crystal by interpreting Demazure operators via elementary bubble
sorting operators acting on weak compositions.
