Gapped Binomial Complexities in Sequences

We relate the gapped k-deck problem introduced by Golm et al. (ISIT 2022) to notions arising in the literature of combinatorics on words. We consider the complexity functions of infinite sequences that count the number of factors up to the equivalence relation of strings having equal gapped k-decks. We show that the Thue–Morse sequence, the fixed point of the substitution 0↦01, 1↦10, has unbounded 1-gap k-binomial complexity for k ≥2. We also show that for a Sturmian sequence and g ≥1, all of its long enough factors are always pairwise g-gap k-binomially inequivalent for any k ≥2.