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''.<ref>{{citation|title=Combinatorics of Permutations|volume=72|series=Discrete Mathematics and Its Applications|first=Miklós|last=Bóna| author-link = Miklós Bóna | edition=2nd|publisher=CRC Press|year=2012|page=227|url=https://books.google.com/books?id=Op-nF-mBR7YC&pg=PA227|isbn=9781439850510}}.</ref>

==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''&nbsp;=&nbsp;''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:

{| class="wikitable" |- ! 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== {{harvs|last=Arratia|authorlink=Richard Arratia|year=1999|txt}} introduced the problem of determining the length of the shortest possible ''k''-superpattern.<ref name="a99">{{citation | last = Arratia | first = Richard | author-link = Richard Arratia | journal = Electronic Journal of Combinatorics | mr = 1710623 | article-number = N1 | title = On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern | url = http://www.combinatorics.org/Volume_6/Abstracts/v6i1n1.html | volume = 6 | year = 1999| doi = 10.37236/1477 | doi-access = free }}</ref> 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 <math>\tbinom{n}{k}\ge k!</math>, from which it follows by Stirling's approximation that ''n''&nbsp;≥&nbsp;''k''²/''e''², where ''e''&nbsp;≈&nbsp;2.71828 is Euler's number. This lower bound was later improved very slightly by {{harvs|last1=Chroman|last2=Kwan|last3=Singhal|year=2021|txt}}, who increased it to 1.000076''k''²/''e''²,<ref name="CKS">{{citation | last1 = Chroman | first1 = Zachary | last2 = Kwan | first2 = Matthew | last3 = Singhal | first3 = Mihir | arxiv = 2004.02375 | doi = 10.1016/j.jcta.2021.105467 | journal = Journal of Combinatorial Theory | mr = 4253319 | at = Paper No. 105467 (15 pp) | series = Series A | title = Lower bounds for superpatterns and universal sequences | volume = 182 | year = 2021 | article-number = 105467 }}</ref> disproving Arratia's conjecture that the ''k''²/''e''² lower bound was tight.<ref name="a99"/>

The upper bound of ''k''² on superpattern length proven by Arratia is not tight. After intermediate improvements,<ref name="eelw">{{citation | last1 = Eriksson | first1 = Henrik | last2 = Eriksson | first2 = Kimmo | last3 = Linusson | first3 = Svante | last4 = Wästlund | first4 = Johan | doi = 10.1007/s00026-007-0329-7 | issue = 3–4 | journal = Annals of Combinatorics | mr = 2376116 | pages = 459–470 | title = Dense packing of patterns in a permutation | volume = 11 | year = 2007| s2cid = 2021533 }}</ref> {{harvs|last=Miller|authorlink=Alison Miller|year=2009|txt}} proved that there is a ''k''-superpattern of length at most ''k''(''k''&nbsp;+&nbsp;1)/2 for every ''k''.<ref name="miller09">{{citation | last = Miller | first = Alison | author-link = Alison Miller | doi = 10.1016/j.jcta.2008.04.007 | issue = 1 | journal = Journal of Combinatorial Theory | pages = 92–108 | series = Series A | title = Asymptotic bounds for permutations containing many different patterns | volume = 116 | year = 2009| doi-access = }}</ref> This bound was later improved by {{harvs|last1=Engen|last2=Vatter|year=2021|txt}}, who lowered it to ⌈(''k''²&nbsp;+&nbsp;1)/2⌉.<ref name="engenvatter">{{citation | last1 = Engen | first1 = Michael | last2 = Vatter | first2 = Vincent | title = Containing all permutations | doi = 10.1080/00029890.2021.1835384 | doi-access = free | journal = American Mathematical Monthly | year = 2021 | volume = 128 | issue = 1 | pages = 4–24 | arxiv = 1810.08252 }}</ref>

Eriksson et al. conjectured that the true length of the shortest ''k''-superpattern is asymptotic to ''k''²/2.<ref name="eelw"/> 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.<ref>{{citation | last1 = Godbole | first1 = Anant P. | last2 = Liendo | first2 = Martha | arxiv = 1302.4668 | doi = 10.1007/s11009-015-9439-6 | issue = 2 | journal = Methodology and Computing in Applied Probability | mr = 3488590 | pages = 517–528 | title = Waiting time distribution for the emergence of superpatterns | volume = 18 | year = 2016}}</ref> {{harvtxt|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 ''k''²/4 to have high probability of being a ''k''-superpattern: permutations shorter than this will likely not contain the identity pattern.<ref name="a99"/> He attributes to Alon the conjecture that, for any {{nowrap|ε > 0}}, with high probability, random permutations of length {{nowrap|''k''²/(4 &minus; ε)}} will be ''k''-superpatterns.

==See also== * Superpermutation

==References== {{reflist}}

Category:Permutation patterns