Computing abelian complexity of binary uniform morphic words

Although a lot of research has been done on the factor complexity (also called subword complexity) of morphic words obtained as fixed points of iterated morphisms, there has been little development in exploring algorithms that can efficiently compute their abelian complexity. The factor complexity counts the number of factors of a given length n  , while the abelian complexity counts that number up to letter permutation. We propose and analyze a simple O(n)O(n) algorithm for quickly computing the exact abelian complexities for all indices from 1 up to n  , when considering binary uniform morphisms. Using our algorithm we also analyze the structure in the abelian complexity for that class of morphisms. Our main result implies, in particular, that the infinite word over the alphabet {−1,0,1}{−1,0,1} constructed from the consecutive forward differences of the abelian complexity of a fixed point of a binary uniform morphism is in fact an automatic sequence with the same morphic length. Since the proof produces morphisms that typically contain many redundant letters, we present an efficient algorithm to eliminate them in order to simplify the morphisms and to see the patterns produced more clearly.