Superpattern

In the mathematical study of permutations and permutation patterns, a superpattern 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

Richard 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 $\tbinom{n}{k}\ge k!,$ from which it follows by Stirling's approximation that n ≥ k²/e², where e ≈ 2.71828 is Euler's number. This remains the strongest known lower bound on the length of superpatterns.

However, the upper bound of k² on superpattern length proven by Arratia is not tight. After intermediate improvements,^[3] the best current upper bound on the size of a superpattern is by Alison Miller (2009), who proved that, for every k, there exists a k-superpattern of length at most k(k + 1)/2.^[4]

Eriksson et al. conjectured that the true length of the shortest k-superpattern is within lower-order terms of k²/2, matched on one side by Miller's upper bound.^[3][4] However, this is in contradiction with a conjecture of Richard Arratia that the k²/e² lower bound is tight,^[2] and also contradicts a conjecture of Noga 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.^[5] 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 Noga 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, MR 1710623.
3. 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.
4. 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.
5. Godbole, Anant; Liendo, Martha (2013), Waiting time distribution for the emergence of superpatterns, arXiv:1302.4668.