Yesterday I came across this interesting result dealing with shortest number with the digits 1 through 7 arranged in all possible orders, thanks to John Baez on twitter [1]. Great to discover this here.
De Brujin sequences as mentioned in one of the comments deals with the generalized case involving repetition as well. There is an interesting mnemonic for remembering the composition of different "ganas" (syllabic units) in Sanskrit poetry that is an example of De Brujin sequence - Ya-Maa-Taa-Raa-Ja-Bhaa-Na-Sa-La-Gam . In Sanskrit metre, syllables can be either long (Maa for example) or short (Ma). The basic units are composed of 3 syllables and thus 2x2x2 = 8 possible types - long-short-long for example. These basis units are called ganas and have a name attached to them. The mnemonic helps one decode the composition of a gana. If you want to know the composition of "Ja Gana" for example, go to the part that starts with Ja and get the substring with three syllables starting with Ja i.e Ja-Bhaa-Na. Since these syllables are short, long and short, Ja Gana corresponds to short, long and short.
De Brujin sequences as mentioned in one of the comments deals with the generalized case involving repetition as well. There is an interesting mnemonic for remembering the composition of different "ganas" (syllabic units) in Sanskrit poetry that is an example of De Brujin sequence - Ya-Maa-Taa-Raa-Ja-Bhaa-Na-Sa-La-Gam . In Sanskrit metre, syllables can be either long (Maa for example) or short (Ma). The basic units are composed of 3 syllables and thus 2x2x2 = 8 possible types - long-short-long for example. These basis units are called ganas and have a name attached to them. The mnemonic helps one decode the composition of a gana. If you want to know the composition of "Ja Gana" for example, go to the part that starts with Ja and get the substring with three syllables starting with Ja i.e Ja-Bhaa-Na. Since these syllables are short, long and short, Ja Gana corresponds to short, long and short.
[1] https://twitter.com/johncarlosbaez/status/105338502434972057...
Edit: Formatting and correction