Editing Weak ordering (section)

===Ordered partitions===

A [[partition of a set]] <math>S</math> is a family of non-empty disjoint subsets of <math>S</math> that have <math>S</math> as their union. A partition, together with a [[total order]] on the sets of the partition, gives a structure called by [[Richard P. Stanley]] an '''ordered partition'''<ref>{{citation|first=Richard P.|last=Stanley|authorlink=Richard P. Stanley|title=Enumerative Combinatorics, Vol. 2|series=Cambridge Studies in Advanced Mathematics|volume=62|page=297|publisher=Cambridge University Press|year=1997}}.</ref> and by [[Theodore Motzkin]] a '''list of sets'''.<ref>{{citation|last=Motzkin|first=Theodore S.|authorlink=Theodore Motzkin|contribution=Sorting numbers for cylinders and other classification numbers|location=Providence, R.I.|mr=0332508|pages=167–176|publisher=Amer. Math. Soc.|title=Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968)|year=1971}}.</ref> An ordered partition of a finite set may be written as a [[finite sequence]] of the sets in the partition: for instance, the three ordered partitions of the set <math>\{a, b\}</math> are
<math display=block>\{a\}, \{b\},</math>
<math display=block>\{b\}, \{a\}, \; \text{ and }</math>
<math display=block>\{a, b\}.</math>

In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the partition, and otherwise inherit the order of the sets that contain them.