Multi-Shift de Bruijn Sequence

A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to a multi-shift setting: a m-shift de Bruijn sequence of order n is a word such that every word of length n appears exactly once in w as a factor that starts at an index im+ 1 for some integer i ≥ 0. We show the number of the m-shift de Bruijn sequences of order n is a!a n −1) for 1 ≤ n ≤ m and is (am!)a n−m for 1 ≤ m ≤ n, where a is the size of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important in solving the Frobenius problem in a free monoid.