Tight Bounds for the Number of Absent Scattered Factors

Abstract: A scattered factor of a word $w$ is a word $u$ that can be obtained by deleting arbitary letters from $w$ and keep the order of the remaining. Barker et al. introduced the notion of $k$-universality, calling a word $k$-universal, if it contains all possible words of length $k$ over a given alphabet $\Sigma$ as a scattered factor. Kosche et al. introduced the notion of absent scattered factors to categorise the words not being scattered factors of a given word. In this paper, we investigate tight bounds on the possible number of absent scattered factors of a given length $k$ (also strictly longer than the shortest absent scattered factors) among all words with the same universality extending the results of Kosche et al. Specifically, given a length $k$ and universality index $\iota$, we characterize $\iota$-universal words with both the maximal and minimal number of absent scattered factors of length $k$. For the lower bound, we provide the exact number in a closed form. For the upper bound, we offer efficient algorithms to compute the number based on the constructed words. Moreover, by combining old results, we present an enumeration with constant delay of the set of scattered factors of a fixed length in time $O(|\Sigma||w|)$.