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 substrings of length n on A; de Bruijn sequences are therefore optimally short. There are distinct de Bruijn sequences B(k, n).
The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn. According to him, 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 in 1894, whereas the generalization to larger alphabets is originally due to Tanja van Aardenne-Ehrenfest and himself.
In most applications, A = {0,1}.
The earliest known example of a de Bruijn sequence comes from Sanskrit prosody where, since the work of Pingala, each possible three-syllable pattern of long and short syllables is given a name, such as 'y' for short–long–long and 'm' for long–long–long. To remember these names, the mnemonic yamātārājabhānasalagām is used, in which each three-syllable pattern occurs starting at its name: 'yamātā' has a short–long–long pattern, 'mātārā' has a long–long–long pattern, and so on, until 'salagām' which has a short–short–long pattern. This mnemonic, equivalent to a de Bruijn sequence on binary 3-tuples, is of unknown antiquity, but is at least as old as C. P. Brown's 1869 book on Sanskrit prosody that mentions it and considers it "an ancient line, written by Pāṇini".
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