Here is the picture associated with dual evacuation (with an svg version):
This time the skew tableau t is depicted by the top row, and next rows, downward, result from successive dual evacuations of elements of t, selected in decreasing order.
The local commutativity rules enable us to build the whole family of diagrams, from the top row and from the diagonal, where all partitions are equal to the outer shape of t. The first column depicts the tableau ε*(t), and symmetry between rows and columns yields a 2d-proof that dual evacuation ε* is an involution.
This dual version of the preceding post would be of little interest, if it was not a preamble to a much nicer formula involving promotion. This operator ∂ is defined like evacuation of the minimal element, except that we put in the cell freed by this evacuation, a new entry larger than any other. The following picture shows p successive applications of ∂ to a tableau t (bottom row), where p is the size of t, i.e., the number of entries:
(here is an svg version).
Because the local rules that allow us to build the picture from the bottom row and from one of the diagonals are always the same, we discover that the left half of the picture, bordered by the central column, is exactly the same as the picture associated to evacuation, published in the preceding post; and on the right half of the picture we recognize a dual evacuation, applied to the central column (beware that diagrams in this part of the picture are different from those located at the start of this post, because top row is different). Hence we get a very nice 2d-proof of the following formula, again due to Schützenberger:
∂ p = ε ε*
where the operators are composed in the "right" order, i.e. first ε then ε*. The tableau ε(t) appears as the central column on this picture.