Abelian Complexity and Synchronization
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We present a general method for computing the abelian complexity ρ^ ab_ s (n) of an automatic sequence s in the case where (a) ρ^ ab_ s (n) is bounded by a constant and (b) the Parikh vectors of the length-n prefixes of s form a synchronized sequence. We illustrate the idea in detail, using the free software Walnut to compute the abelian complexity of the Tribonacci word TR = 0102010⋯, the fixed point of the morphism 0 → 01, 1 → 02, 2 → 0. Previously, Richomme, Saari, and Zamboni showed that the abelian complexity of this word lies in { 3,4,5,6,7 }, and Turek gave a Tribonacci automaton computing it. We are able to "automatically" rederive these results, and more, using the method presented here.
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