The Bruhat order is a certain partial ordering of the
elements of the symmetric group S_{n}. It was shown by Lascoux and
Schützenberger that the number of elements of its MacNeille
completion is the number of n × n alternating sign matrices. We
will discuss this result and some further properties of Bruhat order,
including: (1) its connection with the Bruhat decomposition of the
complete flag variety, (2) topological properties based on
lexicographic shellability, and (3) a weighted enumeration of the
maximal chains of Bruhat order based on the theory of Schubert
polynomials. No prior knowledge of Bruhat order, MacNeille completion,
flag varieties, lexicographic shellability, or Schubert polynomials
will be assumed.
