Editing Weak ordering (section)

===Combinatorial enumeration===
{{main|ordered Bell number}}

The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an <math>n</math>-element set is given by the following sequence {{OEIS|id=A000670}}:
{{number of relations}}

These numbers are also called the '''Fubini numbers''' or '''ordered Bell numbers'''.

For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one [[Singleton (mathematics)|singleton]] set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.