  Motivated by the enumeration of a class of plane partitions studied by
Proctor and by considerations about symmetry classes of plane partitions, we
consider the problem of enumerating lozenge tilings of a hexagon with ``maximal
staircases'' removed from some of its vertices. The case of one vertex
corresponds to Proctor's problem. For two vertices there are several cases to
consider, and most of them lead to nice enumeration formulas. For three or more
vertices there do not seem to exist nice product formulas in general, but in
one special situation a lot of factorization occurs, and we pose the problem of
finding a formula for the number of tilings in this case.
