Superpermutation

In combinatorial mathematics, a superpermutation on n symbols is a string that contains each permutation of n symbols as a substring.

It has been shown that for 1 ≤ n ≤ 5, the smallest superpermutation on n symbols has length 1! + 2! + … + n! (sequence A180632 in the OEIS). The first five superpermutations have respective lengths 1, 3, 9, 33, and 153, forming the strings 1, 121, 123121321, 123412314231243121342132413214321, and the string:

123451234152341253412354123145231425314235142315423124531243
512431524312543121345213425134215342135421324513241532413524
132541321453214352143251432154321

The smallest superpermutations found to date for n >= 6 have length (1! + 2! + ... + n!) - 1. These superpermutations come from a construction that takes a superpermutation on n - 1 symbols of length k and produces a superpermutation on n symbols of length k + n!.[1]

See also

Further reading

  • Ashlock, Daniel A.; Tillotson, Jenett (1993), "Construction of small superpermutations and minimal injective superstrings", Congressus Numerantium, 93: 91–98, Zbl 0801.05004
  • Johnston, Nathaniel (July 28, 2013), "Non-uniqueness of minimal superpermutations", Discrete Mathematics, 313 (14): 1553–1557, arXiv:1303.4150, doi:10.1016/j.disc.2013.03.024, Zbl 1368.05004, retrieved March 16, 2014
  • Houston, Robin (21 August 2014), Tackling the Minimal Superpermutation Problem, arXiv:1408.5108, Bibcode:2014arXiv1408.5108H

References

  1. ↑ Robin Houston (2014). "Tackling the Minimal Superpermutation Problem" (PDF).
  • The Minimal Superpermutation Problem - Nathaniel Johnston's blog
  • Grime, James. "Superpermutations - Numberphile" (video). YouTube. Brady Haran. Retrieved 1 February 2018.


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