On scattered subword complexity

Sequences of characters called words or strings are widely studied in combinatorics, and used in various fields of sciences (e.g. chemistry, physics, social sciences, biology [2, 3, 4, 11] etc.). The elements of a word are called letters. A contiguous part of a word (obtained by erasing a prefix or/and a suffix) is a subword or factor. If we erase arbitrary letters from a word, what is obtained is a scattered subword. Special scattered subwords, in which the consecutive letters are at distance at most d (d ≥ 1) in the original word, are called d-subwords [7, 8]. In [9] the super -d-subword is defined, in which case the distances are of length at least d. The super-d-complexity, as the number of such subwords, is computed for rainbow words (words with pairwise different letters). In this paper we define special scattered subwords, for which the distance in the original word of length n between two letters which will be consecutive in the subword, is taken from a subset of {1, 2, . . . , n − 1}. The complexity of a word is defined as the number of all its different subwords. Similar definitions are for d-complexity, super-d-complexity and scattered subword complexity.