Permutation complexity of the Thue–Morse word

Given a countable set X (usually taken to be N or Z), an infinite permutation π of X is a linear ordering ≺π of X, introduced in Fon-Der-Flaass and Frid (2007) [5]. This paper investigates the combinatorial complexity of the infinite permutation on N 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.