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, ..., s_k 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:

Subsequence	Pattern
253	132
251	231
254	132
231	231
234	123
214	213
531	321
534	312
514	312
314	213

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 ${\displaystyle {\tbinom {n}{k}}\geq k!}$, 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 (2020), 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 (2019), 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

• Superpermutation

References

1. Bóna, Miklós (2012), Combinatorics of Permutations, Discrete Mathematics and Its Applications, 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 (2020), Lower bounds for superpatterns and universal sequences, arXiv:2004.02375
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
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, doi:10.1080/00029890.2021.1835384
7. Godbole, Anant; Liendo, Martha (2013), Waiting time distribution for the emergence of superpatterns, arXiv:1302.4668, Bibcode:2013arXiv1302.4668G.