On the shape of subword complexity sequences of finite words

If u and v are words, uv denotes their concatenation. For a positive integer n, u = uuu...u (n times). By ǫ we denote the only word of length 0, the empty word. A is the set of all words over A of length n. A = ∪n≥1A n is a free semigroup (an associative set with a binary operation) over A with the group operation being string concatenation. We define A = A ∪ {ǫ}. Definition 1.2. A word u is a subword of a word w if there exist p, q ∈ A such that w = puq. Equivalently, we say that a word u is a subword (factor) of a word w = w1w2 . . . wn if there exist integers i, j ∈ N with 0 < i ≤ j such that u = wiwi+1 . . . wj. We denote this occurrence of u in w by w[i, j].