Superpattern

In the mathematical study of permutations and permutation patterns, a superpattern or universal permutation is a permutation that contains all of the patterns of a given length. More specifically, a k-superpattern contains all possible patterns of length k.[1]

Definitions and example

If π is a permutation of length n, represented as a sequence of the numbers from 1 to n in some order, and s = s1, s2, ..., sk is a subsequence of π of length k, then s corresponds to a unique pattern, a permutation of length k whose elements are in the same order as s. That is, for each pair i and j of indexes, the ith element of the pattern for s should be less than the jthe element if and only if the ith element of s is less than the jth element. Equivalently, the pattern is order-isomorphic to the subsequence. For instance, if π is the permutation 25314, then it has ten subsequences of length three, forming the following patterns:

SubsequencePattern
253132
251231
254132
231231
234123
214213
531321
534312
514312
314213

A permutation π is called a k-superpattern if its patterns of length k include all of the length-k permutations. For instance, the length-3 patterns of 25314 include all six of the length-3 permutations, so 25314 is a 3-superpattern. No 3-superpattern can be shorter, because any two subsequences that form the two patterns 123 and 321 can only intersect in a single position, so five symbols are required just to cover these two patterns.

Length bounds

Arratia (1999) introduced the problem of determining the length of the shortest possible k-superpattern.[2] He observed that there exists a superpattern of length k2 (given by the lexicographic ordering on the coordinate vectors of points in a square grid) and also observed that, for a superpattern of length n, it must be the case that it has at least as many subsequences as there are patterns. That is, it must be true that , from which it follows by Stirling's approximation that n  k2/e2, where e  2.71828 is Euler's number. This lower bound was later improved very slightly by Chroman, Kwan, and Singhal (2021), who increased it to 1.000076k2/e2,[3] disproving Arratia's conjecture that the k2/e2 lower bound was tight.[2]

The upper bound of k2 on superpattern length proven by Arratia is not tight. After intermediate improvements,[4] Miller (2009) proved that there is a k-superpattern of length at most k(k + 1)/2 for every k.[5] This bound was later improved by Engen and Vatter (2021), who lowered it to ⌈(k2 + 1)/2⌉.[6]

Eriksson et al. conjectured that the true length of the shortest k-superpattern is asymptotic to k2/2.[4] However, this is in contradiction with a conjecture of Alon on random superpatterns described below.

Random superpatterns

Researchers have also studied the length needed for a sequence generated by a random process to become a superpattern.[7] Arratia (1999) observes that, because the longest increasing subsequence of a random permutation has length (with high probability) approximately 2√n, it follows that a random permutation must have length at least k2/4 to have high probability of being a k-superpattern: permutations shorter than this will likely not contain the identity pattern.[2] He attributes to Alon the conjecture that, for any ε > 0, with high probability, random permutations of length k2/(4 ε) will be k-superpatterns.

See also

References

  1. Bóna, Miklós (2012), Combinatorics of Permutations, Discrete Mathematics and Its Applications, vol. 72 (2nd ed.), CRC Press, p. 227, ISBN 9781439850510.
  2. Arratia, Richard (1999), "On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern", Electronic Journal of Combinatorics, 6: N1, doi:10.37236/1477, MR 1710623
  3. Chroman, Zachary; Kwan, Matthew; Singhal, Mihir (2021), "Lower bounds for superpatterns and universal sequences", Journal of Combinatorial Theory, Series A, 182, Paper No. 105467 (15 pp), arXiv:2004.02375, doi:10.1016/j.jcta.2021.105467, MR 4253319
  4. Eriksson, Henrik; Eriksson, Kimmo; Linusson, Svante; Wästlund, Johan (2007), "Dense packing of patterns in a permutation", Annals of Combinatorics, 11 (3–4): 459–470, doi:10.1007/s00026-007-0329-7, MR 2376116, S2CID 2021533
  5. Miller, Alison (2009), "Asymptotic bounds for permutations containing many different patterns", Journal of Combinatorial Theory, Series A, 116 (1): 92–108, doi:10.1016/j.jcta.2008.04.007
  6. Engen, Michael; Vatter, Vincent (2021), "Containing all permutations", American Mathematical Monthly, 128 (1): 4–24, arXiv:1810.08252, doi:10.1080/00029890.2021.1835384
  7. Godbole, Anant P.; Liendo, Martha (2016), "Waiting time distribution for the emergence of superpatterns", Methodology and Computing in Applied Probability, 18 (2): 517–528, arXiv:1302.4668, doi:10.1007/s11009-015-9439-6, MR 3488590
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