Arithmetical Complexity of Infinite Words

Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words

“…If b (2) x (n) = n + 1, for all n 0, then the factor complexity function p x is unbounded and x is aperiodic. As a consequence of Theorem 2, an infinite word x is Sturmian if and only if, for all n 1 and all m 2, b (1) x (n) = 2 and b (m)…”

“…Note that b (1) x corresponds to the usual abelian complexity denoted by ρ ab x . If p x denotes the usual factor complexity, then for all m 1, we have…”

“…First note that Sturmian words have a constant abelian complexity. Hence, if x is a Sturmian word, then b (1) x (n) = 2 for all n 1.…”

“…If x is a right-infinite word such that b (1) x (n) = 2 for all n 1, then x is clearly balanced. If b (2) x (n) = n + 1, for all n 0, then the factor complexity function p x is unbounded and x is aperiodic.…”

“…Already in 1961, Erdős opened the way to a new research direction by raising the question of avoiding abelian squares in arbitrarily long words [6]. Related to Van der Waerden theorem, we can also mention the arithmetic complexity [1] mapping n 0 to the number of distinct subwords x i x i+p · · · x i+(n−1)p built from n letters arranged in arithmetic progressions in the infinite word x, i 0, p 1. In the same direction, one can also consider maximal pattern complexity [7].…”