A conjecture on the 2-abelian complexity of the Thue-Morse word

The Thue-Morse word is a well-known and extensively studied 2-automatic
sequence. For example, it is trivially abelian periodic and its abelian
complexity takes only two values. For an integer k, the k-abelian
complexity is a generalization of the abelian complexity, corresponding
to the case where k=1.

Formally, two words u and v of the same length are k-abelian equivalent
if they have the same prefix (resp. suffix) of length k-1 and if, for
all words x of length k, the numbers of occurrences of x in u and v are
the same. This notion has received some recent interest, see the works
of Karhumäki et al. The k-abelian complexity of an infinite word x maps
an integer n to the number of k-abelian classes partitioning the set of
factors of length n occurring in x.

The aim of this talk is to study the 2-abelian complexity a(n) of the
Thue-Morse word. We conjecture that a(n) is 2-regular in the sense of
Allouche and Shallit. This question can be related to a work of Madill
and Rampersad (2012) where the (1)-abelian complexity of the paper
folding word is shown to be 2-regular. We will present some arguments
supporting our conjecture. They are based on functions counting some
subword of length 2 occuring in prefixes of the Thue-Morse word.