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''.

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'' = ''s''₁, ''s''₂, ..., ''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 ''i''-th element of the pattern for ''s'' should be less than the ''j''-th element if and only if the ''i''-th element of ''s'' is less than the ''j''-th 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

introduced the problem of determining the length of the shortest possible ''k''-superpattern. He observed that there exists a superpattern of length ''k''² (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 $\tbinom\ge k!$, from which it follows by Stirling's approximation that ''n'' ≥ ''k''²/''e''², where ''e'' ≈ 2.71828 is Euler's number.
This lower bound was later improved very slightly by
,
who increased it to 1.000076''k''²/''e''², disproving Arratia's conjecture that the ''k''²/''e''² lower bound was tight.

The upper bound of ''k''² on superpattern length proven by Arratia is not tight. After intermediate improvements,
proved that there is a ''k''-superpattern of length at most ''k''(''k'' + 1)/2 for every ''k''.
This bound was later improved by
, who lowered it to ⌈(''k''² + 1)/2⌉.

Eriksson et al. conjectured that the true length of the shortest ''k''-superpattern is asymptotic to ''k''²/2.
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. 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 ''k''²/4 to have high probability of being a ''k''-superpattern: permutations shorter than this will likely not contain the identity pattern. He attributes to Alon the conjecture that, for any , with high probability, random permutations of length will be ''k''-superpatterns.

See also

* Superpermutation

References

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Permutation patterns