Step 3:

Determine a complete list of all the elements of a discrete structure. In a complete list of a discrete structure each element of this structure occurs exactly once.

Step 4:

Generate the objects of a discrete structure uniformly at random. In other words we apply an algorithm which generates each object of a discrete structure with exactly the same probability. Especially in the case when the objects of a discrete structure are themselves classes of (equivalent) objects, then we ask that the algorithm hits each class with the same probability, even if not all the classes are of the same size.

In general, step 3 is the most ambitious task, it needs a lot of computing power, computing time and memory. For that reason, when the set of all elements of a discrete structure is too hard to be completely determined it is useful and makes sense to consider step 4. This approach allows to generate a huge variety of different unprejudiced objects of a discrete structure. These sets of examples can be very useful for checking certain hypotheses on them and afterwards for trying to prove the valid ones.

For example, let us have a short look at the classification of unlabelled graphs on vertices:

Step 1:

There are graphs on vertices.

Step 2:

There exists exactly one graph with edges, with edge, with edges or with edges; two graphs with or edges; three graphs with edges.

Step 3:

The unlabelled graphs on vertices are given in figure 1.

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Figure 1: The unlabelled graphs on vertices.

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In this example step 4 is trivial, whence it is omitted.

As was already mentioned, the standard tool for classification of discrete structures is the notion of group actions. A detailed introduction to combinatorics under finite group actions can be found in Kerber (1991, 1999).

A multiplicative group with neutral element acts (from the left) on a set if there exists a mapping