DNA-words and word posets

In the paper two variants of a combinatorial problem for the set F q of sequences of length n over the alphabet Fq = {0, 1, .., q − 1} are considered, with some applications. The original problem was the following: what is the smallest k such that every word v ∈ F q is uniquely determined by the set of its subwords of length up to k. This problem was solved by Lothaire [1]. We consider the following variant of this problem: the n-letter word w = w1...wn (which is called a DNA-word) is composed over an alphabet consisting of q complement pairs:{i, ī : i = 0, .., q− 1}; and denote by w its reverse complement, i.e. w = w̄n...w̄1. A DNA-word u is called a subword of w if it is a subword of either w or w. As above, we’re looking for the smallest k. We give a simple proof for k = n − 1, and apply this result for determining the automorphism group of the poset of DNA-words of length at most n, partially ordered by the above subword relation. Furthermore we give a sharp result k ∼ 2n/3, which is an analogue of the former result [1].