Binary sequences | Enumerative combinatorics
De Bruijn sequence
In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a contiguous subsequence). Such a sequence is denoted by B(k, n) and has length kn, which is also the number of distinct strings of length n on A. Each of these distinct strings, when taken as a substring of B(k, n), must start at a different position, because substrings starting at the same position are not distinct. Therefore, B(k, n) must have at least kn symbols. And since B(k, n) has exactly kn symbols, De Bruijn sequences are optimally short with respect to the property of containing every string of length n at least once. The number of distinct de Bruijn sequences B(k, n) is The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn, who wrote about them in 1946. As he later wrote, the existence of de Bruijn sequences for each order together with the above properties were first proved, for the case of alphabets with two elements, by Camille Flye Sainte-Marie. The generalization to larger alphabets is due to Tatyana van Aardenne-Ehrenfest and de Bruijn. Automata for recognizing these sequences are denoted as de Bruijn automata. In most applications, A = {0,1}. (Wikipedia).
