  Given a countable set X (usually taken to be the natural numbers or the
integers), an infinite permutation \pi of X is a linear ordering of X. This
paper investigates the combinatorial complexity of the infinite permutation on
the natural numbers associated with the well-known and well-studied Thue-Morse
word. A formula for the complexity is established by studying patterns in
subpermutations and the action of the Thue-Morse morphism on the
subpermutations.
