Gray codes for noncrossing and nonnesting partitions of classical types

One of the fundamental topics on the area of combinatorial algorithms is the efficient generation of all objects in a specific combinatorial class in such a way that each item is generated exactly once, hence producing a listing of all objects in the considered class. A common approach to this problem has been the generation of the objects of a combinatorial class in such a way that two consecutive items differ in some pre-specified, usually small, way. Such generation is usually called a Gray code and, amongst the various applications of combination generation, Gray codes are especially valued since they usually involve recursive constructions which provide new insights into the structure of the combinatorial class [11]. The problem of finding a Gray code for a combinatorial class can be formulated as a Hamilton path/cycle problem: the vertices of the graph are the objects themselves, and two vertices are joined by an edge if they differ in a pre-specified way. This graph has a Hamilton path if and only if the required listing of the objects