What I mean is that the operation changes the part in place to yield a part that is strictly smaller than the original, but is larger than any other such smaller part.  If these were integers, the operation would be "subtract 1", but of course partitions are a bit more complicated.

How about:
# Decrement - change the part so that it is smaller in the 
#   (vector) lexicographic order, but reduced by the smallest amount possible.

For the corresponding method in MultisetPartitionTraverser I use the phrase

        If you think of the v values in the part as a multi-digit
        integer (least significant digit on the right) this is
        basically decrementing that integer, but with the extra
        constraint that the leftmost digit cannot be decremented to 0.

Would you find that any clearer?  (Even the method comment is a little bit over-simplified, because the "integer" in question is one in which each digit is in a different base.)

It is important to make this clear, because the variants decrement_part_{small|large|range} in the class are the core of the constrained enumerate algorithms.  (And those are quite a but more complicated.)