Superpattern

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:

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. 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 ( n k ) ≥ 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, 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.

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. However, this is in contradiction with a conjecture of Richard Arratia that the k²/e² lower bound is tight, 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. 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. 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.