Abelian complexity of fixed point of morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1

Abstract

We study the combinatorics of vtm, a variant of the Thue-Morse word generated by the non-uniform morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1 starting with 0. This infinite ternary sequence appears a lot in the literature and finds applications in several fields such as combinatorics on words; for example, in pattern avoidance it is often used to construct infinite words avoiding given patterns. It has been shown that the factor complexity of vtm, i.e., the number of factors of length n, is Θ(n); in fact, it is bounded by ¹⁰⁄₃n for all n, and it reaches that bound precisely when n can be written as 3 times a power of 2. In this paper, we show that the abelian complexity of vtm, i.e., the number of Parikh vectors of length n, is O(log n) with constant approaching ¾ (assuming base 2 logarithm), and it is Ω(1) with constant 3 (and these are the best possible bounds). We also prove some results regarding factor indices in vtm.