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In [[combinatorics|combinatorial mathematics]] and [[theoretical computer science]], a (classical) '''permutation pattern''' is a sub-permutation of a longer [[permutation]]. Any permutation may be written in [[Permutation#One-line_notation|one-line notation]] as a sequence of entries representing the result of applying the permutation to the sequence 123...; for instance the sequence 213 represents the permutation on three elements that swaps elements 1 and 2. If  π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to the number [[pi]]), then π is said to ''contain'' σ as a ''pattern'' if some [[subsequence]] of the entries of π has the same relative order as all of the entries of σ.

For instance, permutation π contains the pattern 213 whenever π has three entries ''x'', ''y'', and ''z'' that appear within π in the order ''x''...''y''...''z'' but whose values are ordered as ''y''&nbsp;<&nbsp;''x''&nbsp;<&nbsp;''z'', the same as the ordering of the values in the permutation 213. 

The permutation 32415 on five elements contains 213 as a pattern in several different ways: 3··15, ··415, 32··5, 324··, and ·2·15 all form triples of entries with the same ordering as 213. Note that the entries do not need to be consecutive. Each of the subsequences 315, 415, 325, 324, and 215 is called a ''copy,'' ''instance,'' or ''occurrence'' of the pattern. The fact that π contains σ is written more concisely as σ &le; π. 

If a permutation π does not contain a pattern σ, then π is said to ''avoid'' σ. The permutation 51342 avoids 213; it has ten subsequences of three entries, but none of these ten subsequences has the same ordering as 213.

== Early results ==
A case can be made that {{harvs|first=Percy|last=MacMahon|authorlink=Percy MacMahon|year=1915|txt}} was the first to prove a result in the field with his study of "lattice permutations".<ref>{{ Citation
 | last=MacMahon
 | first=Percy A.
 | author-link=Percy Alexander MacMahon
 | title=Combinatory Analysis
 | publisher=Cambridge University Press
 | place=London
 | year=1915
 | at = [https://archive.org/details/combinatoryanal01macmuoft/page/124 Volume I, Section III, Chapter V]
}}.</ref>  In particular MacMahon shows that the permutations which can be divided into two decreasing subsequences (i.e., the 123-avoiding permutations) are counted by the [[Catalan number]]s.<ref>{{harvtxt|MacMahon|1915}}, Items 97 and 98.</ref>

Another early landmark result in the field is the [[Erdős–Szekeres theorem#Permutation pattern interpretation|Erdős–Szekeres theorem]]; in permutation pattern language, the theorem states that for any positive integers ''a'' and ''b'' every permutation of length at least  <math>(a-1)(b-1) + 1 </math> must contain either the pattern <math>1, 2, 3,\dots , a  </math> or the pattern <math>b,b-1,\dots , 2, 1</math>.

== Computer science origins ==
The study of permutation patterns began in earnest with [[Donald Knuth]]'s consideration of [[Stack-sortable permutation|stack-sorting]] in 1968.<ref>{{ Citation
 | last=Knuth
 | first=Donald E.
 | author-link=Donald Knuth
 | title=[[The Art of Computer Programming|The Art Of Computer Programming Vol. 1]]
 | publisher=Addison-Wesley
 | place=Boston
 | year=1968
 | isbn=0-201-89683-4
 | oclc=155842391
 | mr= 0286317}}.</ref>  Knuth showed that the permutation π can be sorted by a [[Stack (data structure)|stack]] if and only if π avoids 231, and that the stack-sortable permutations are enumerated by the [[Catalan number]]s.<ref>{{harvtxt|Knuth|1968}}, Section 2.2.1, Exercises 4 and 5.</ref>  Knuth also raised questions about sorting with [[Double-ended queue|deques]].  In particular, Knuth's question asking how many permutation of ''n'' elements are obtainable with the use of a deque remains open.<ref>{{harvtxt|Knuth|1968}}, Section 2.2.1, Exercise 13, rated M49 in the first printing, and M48 in the second.</ref>  Shortly thereafter, {{harvs|first=Robert|last=Tarjan|year=1972|authorlink=Robert Tarjan|txt}} investigated sorting by networks of stacks,<ref>{{Citation | last1=Tarjan | first1=Robert | author1-link=Robert Tarjan | title=Sorting using networks of queues and stacks | mr = 0298803 | year=1972 | journal=[[Journal of the ACM]] | volume=19 | issue=2 | pages=341–346 | doi = 10.1145/321694.321704| s2cid=13608929 | doi-access=free }}.</ref> while {{harvs|first=Vaughan|last=Pratt|authorlink=Vaughan Pratt|year=1973|txt}} showed that the permutation π can be sorted by a deque if and only if for all ''k'', π avoids 5,2,7,4,...,4''k''+1,4''k''&minus;2,3,4''k'',1, and 5,2,7,4,...,4''k''+3,4''k'',1,4''k''+2,3, and every permutation that can be obtained from either of these by interchanging the last two elements or the 1 and the 2.<ref name="pratt73">{{Citation | last1=Pratt | first1=Vaughan R. | author1-link=Vaughan Pratt | contribution=Computing permutations with double-ended queues. Parallel stacks and parallel queues |mr = 0489115 | year=1973 | title=[[Symposium on Theory of Computing|Proc. Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973)]] | pages=268–277 | doi = 10.1145/800125.804058| s2cid=15740957 | doi-access=free }}.</ref>  Because this collection of permutations is infinite (in fact, it is the first published example of an infinite [[antichain]] of permutations), it is not immediately clear how long it takes to decide if a permutation can be sorted by a deque.  {{harvtxt|Rosenstiehl|Tarjan|1984}} later presented a linear (in the length of π) time algorithm which determines if π can be sorted by a deque.<ref>{{Citation | last1=Rosenstiehl | first1=Pierre | author1-link=Pierre Rosenstiehl | last2=Tarjan | first2=Robert | author2-link=Robert Tarjan | title=Gauss codes, planar Hamiltonian graphs, and stack-sortable permutations | mr = 756164 | year=1984 | journal=Journal of Algorithms | volume=5 | issue=3 | pages=375–390 | doi = 10.1016/0196-6774(84)90018-X}}.</ref>

In his paper, Pratt remarked that this permutation pattern order “seems to be the only partial order on permutation that arises in a simple and natural way” and concludes by noting that “from an abstract point of view”, the permutation pattern order “is even more interesting than the networks we were characterizing”.<ref name="pratt73"/>

== Enumerative origins ==
{{main article|Enumerations of specific permutation classes}}
A prominent goal in the study of permutation patterns is in the enumeration of permutations avoiding a fixed (and typically short) permutation or set of permutations. Let Av''<sub>n</sub>''(B) denote the set of permutations of length ''n'' which avoid all of the permutations in the set ''B''. (In the case ''B'' is a singleton, say {''β''}, the abbreviation Av''<sub>n</sub>''(''β'') is used instead.) As noted above, MacMahon and Knuth showed that |Av''<sub>n</sub>''(123)| = |Av''<sub>n</sub>''(231)|  = ''C<sub>n</sub>'', the ''n''th Catalan number. Thus these are isomorphic [[combinatorial class]]es.

{{harvtxt|Simion|Schmidt|1985}} was the first paper to focus solely on enumeration.  Among other results, Simion and Schmidt counted [[Parity of a permutation|even and odd permutations]] avoiding a pattern of length three, counted permutations avoiding [[Enumerations of specific permutation classes#Classes avoiding two patterns of length 3|two patterns of length three]], and gave the first bijective proof that 123- and 231-avoiding permutations are equinumerous.<ref>{{Citation
| last1=Simion | first1=Rodica | author1-link = Rodica Simion
| last2=Schmidt | first2=Frank W.
| title=Restricted permutations
| year=1985
| journal=[[European Journal of Combinatorics]]
| volume=6
| issue=4 | pages=383–406
| mr = 0829358
 | doi=10.1016/s0195-6698(85)80052-4
| doi-access=free
}}.</ref>  Since their paper, many other bijections have been given, see {{harvtxt|Claesson|Kitaev|2008}} for a survey.<ref>{{Citation| last1=Claesson | first1=Anders| last2=Kitaev | first2=Sergey| author2-link = Sergey Kitaev| arxiv = 0805.1325| title=Classification of bijections between 321- and 132-avoiding permutations| year=2008| journal=[[Séminaire Lotharingien de Combinatoire]]| url = https://www.mat.univie.ac.at/~slc/wpapers/s60claekit.html| volume=60| pages=B60d, 30pp| mr = 2465405}}.</ref>

In general, if |Av''<sub>n</sub>''(''β'')| = |Av''<sub>n</sub>''(''σ'')| for all ''n'', then ''β'' and ''σ'' are said to be [[Wilf equivalence|''Wilf-equivalent'']].  Many Wilf-equivalences stem from the trivial fact that |Av''<sub>n</sub>''(''β'')| = |Av''<sub>n</sub>''(''β''<sup>&minus;1</sup>)| = |Av''<sub>n</sub>''(''β''<sup>rev</sup>)| for all ''n'', where ''β''<sup>&minus;1</sup> denotes the [[Permutation#Product and inverse|inverse]] of ''β'' and ''β''<sup>rev</sup> denotes the reverse of ''β''.  (These two operations generate the [[Examples of groups#The symmetry group of a square - dihedral group of order 8|Dihedral group D<sub>8</sub>]] with a natural action on [[permutation matrices]].)  However, there are also numerous examples of nontrivial Wilf-equivalences (such as that between ''123'' and ''231''):

* {{harvtxt|Stankova|1994}} proved that the permutations 1342 and 2413 are Wilf-equivalent.<ref>{{Citation | last1=Stankova | first1=Zvezdelina | title=Forbidden subsequences | mr = 1297387 | year=1994 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]] | volume=132 | issue=1–3 | pages=291–316 | doi = 10.1016/0012-365X(94)90242-9| doi-access=free }}.</ref>
* {{harvtxt|Stankova|West|2002}} proved that for any permutation ''β'', the permutations 231 &oplus; ''β'' and 312 &oplus; ''β'' are Wilf-equivalent, where &oplus; denotes the [[direct sum of permutations|direct sum]] operation.<ref>{{Citation | last1=Stankova | first1=Zvezdelina | last2=West | first2=Julian | title=A New class of Wilf-Equivalent Permutations | mr = 1900628 | year=2002 | journal=[[Journal of Algebraic Combinatorics]] | volume=15 | issue=3 | pages=271–290 | doi = 10.1023/A:1015016625432| arxiv=math/0103152 | s2cid=13921676 }}.</ref>
* {{harvtxt|Backelin|West|Xin|2007}} proved that for any permutation ''β'' and any positive integer ''m'', the permutations 12..''m'' &oplus; ''β'' and ''m''...21&nbsp;&oplus;&nbsp;''β'' are Wilf-equivalent.<ref>{{citation | last1=Backelin | first1=Jörgen | last2=West | first2=Julian | last3=Xin | first3=Guoce | title=Wilf-equivalence for singleton classes 
 | mr = 2290807 | year=2007 | journal=[[Advances in Applied Mathematics]] | volume=38 | issue=2 | pages=133–149 | doi = 10.1016/j.aam.2004.11.006| doi-access=free }}.</ref>

From these two Wilf-equivalences and the inverse and reverse symmetries, it follows that there are three different sequences |Av''<sub>n</sub>''(''β'')| where ''β'' is of length four:

{| class="wikitable" style="text-align:center;" border="1"
|-
! ''β'' !! sequence enumerating Av''<sub>n</sub>''(''β'') !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] reference !! exact enumeration reference
|-
| &nbsp;1342&nbsp; || 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, ... || {{OEIS link|id=A022558}} || {{harvtxt|Bóna|1997}}<ref>{{citation | last1=Bóna | first1=Miklós | authorlink = Miklós Bóna |title=Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps | mr = 1485138 | year=1997 | journal=[[Journal of Combinatorial Theory]]|series= Series A | volume=80 | issue=2 | pages=257–272 | doi = 10.1006/jcta.1997.2800| arxiv=math/9702223 | s2cid=18352890 }}.</ref>
|-
| &nbsp;1234&nbsp; || 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, ... || {{OEIS link|id=A005802}} || {{harvtxt|Gessel|1990}}<ref name="gessel90">{{Citation | last1=Gessel | first1=Ira M. | title=Symmetric functions and ''P''-recursiveness | mr = 1041448 | year=1990 | journal=[[Journal of Combinatorial Theory]]|series= Series A | volume=53 | issue=2 | pages=257–285 | doi = 10.1016/0097-3165(90)90060-A| doi-access=free }}.</ref>
|-
| &nbsp;1324&nbsp; || 1, 2, 6, 23, 103, 513, 2762, 15793, 94776, 591950, ... || {{OEIS link|id=A061552}} || unenumerated
|}

In the late 1980s, [[Richard P. Stanley|Richard Stanley]] and [[Herbert Wilf]] conjectured that for every permutation ''β'', there is some constant ''K'' such that |Av''<sub>n</sub>''(''β'')| < ''K<sup>n</sup>''.  This was known as the [[Stanley–Wilf conjecture]] until it was proved by [[Adam Marcus (mathematician)|Adam Marcus]] and [[Gábor Tardos]].<ref>{{Citation | last1=Marcus | first1=Adam | last2=Tardos | first2= Gábor | author2-link=Gábor Tardos | title=Excluded permutation matrices and the Stanley-Wilf conjecture | mr = 2063960 | year=2004 | journal=[[Journal of Combinatorial Theory]]|series=Series A  | volume=107 | issue=1 | pages=153–160 | doi=10.1016/j.jcta.2004.04.002| doi-access=free }}.</ref>

== Closed classes ==
{{main|Permutation class}}
A ''closed class'', also known as a ''pattern class'', ''permutation class'', or simply ''class'' of permutations is a [[Ideal (order theory)|downset]] in the permutation pattern order.  Every class can be defined by the minimal permutations which do not lie inside it, its ''basis''.  Thus the basis for the stack-sortable permutations is {231}, while the basis for the deque-sortable permutations is infinite.  The ''generating function'' for a class is Σ x<sup>|π|</sup> where the sum is taken over all permutations π in the class.

== Möbius function ==
As the set of permutations under the containment order forms a [[Partially ordered set|poset]] it is natural to ask about its [[incidence algebra#Special elements|Möbius function]], a goal first explicitly presented by {{harvtxt|Wilf|2002}}.<ref>{{citation | last1=Wilf | first1=Herbert | authorlink = Herbert Wilf |title=Patterns of permutations | mr = 1935750 | year=2002 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]| volume=257 | issue=2 | pages=575–583 | doi = 10.1016/S0012-365X(02)00515-0| doi-access=free }}.</ref>
The goal in such investigations is to find a formula for the Möbius function of an interval [σ, π] in the permutation pattern poset which is more efficient than the naïve recursive definition. The first such result was established by {{harvtxt|Sagan|Vatter|2006}}, who gave a formula for the Möbius function of an interval of [[layered permutation]]s.<ref>{{Citation | last1=Sagan | first1=Bruce | author1-link=Bruce Sagan | last2=Vatter | first2= Vince | title=The Möbius function of a composition poset | mr = 2259013 | year=2006 | journal=[[Journal of Algebraic Combinatorics]]  | volume=24 | issue=2 | pages=117–136 | doi=10.1007/s10801-006-0017-4| arxiv=math/0507485 | s2cid=11283347 }}.</ref>
Later, {{harvtxt|Burstein|Jelinek|Jelinkova|Steingrimsson|2011}} generalized this result to intervals of [[separable permutation]]s.<ref>{{Citation | last1=Burstein | first1=Alexander | last2=Jelinek | first2= Vit | last3=Jelinkova | first3=Eva | last4=Steingrimsson | first4= Einar |title=The Möbius function of separable and decomposable permutations | mr = 2834180 | year=2011 | journal=[[Journal of Combinatorial Theory]]|series=Series A  | volume=118 | issue=1 | pages=2346–2364 | doi=10.1016/j.jcta.2011.06.002| s2cid=13978488 | doi-access=free }}.</ref>

It is known that, asymptotically, at least 39.95% of all permutations π of length ''n'' satisfy μ(1, π)=0 (that is, the principal Möbius function is equal to zero),<ref>{{citation |last1=Brignall |first1=Robert |last2=Jelínek |first2=Vit |last3=Kynčl |first3=Jan |last4=Marchant |first4=David |title=Zeros of the Möbius function of permutations | year=2019 | journal=[[Mathematika]] |volume=65 |issue=4 |pages=1074–1092 | mr=3992365 | doi=10.1112/S0025579319000251|s2cid=53366318 |url=http://oro.open.ac.uk/66369/1/181012-Marchant-v101.pdf |arxiv=1810.05449 }}</ref> but for each ''n'' there exist permutations π such that μ(1, π) is an exponential function of ''n''.<ref>{{citation |last1=Marchant |first1=David |title=2413-balloon permutations and the growth of the Möbius function |journal=[[Electronic Journal of Combinatorics]] |date=2020 |volume=27 |issue=1 |page=Article P1.7, 18 pp |doi=10.37236/8554|doi-access=free |arxiv=1812.05064 }}</ref>

== Computational complexity ==

Given a permutation <math>\tau</math> (called the ''text'') of length <math>n</math> and another permutation <math>\pi</math> of length <math>k</math> (called the ''pattern''), the ''permutation pattern matching (PPM)'' problem asks whether <math>\pi</math> is contained in <math>\tau</math>. When both <math>n</math> and <math>k</math> are regarded as variables, the problem is known to be [[NP-complete]], and the problem of counting the number of such matches is [[Sharp-P-complete|#P-complete]].<ref name="BBL 1998">{{citation|last1=Bose|first1=Prosenjit|author1-link=Jit Bose|last2=Buss|first2=Jonathan F.|last3=Lubiw|first3=Anna|author3-link=Anna Lubiw|title=Pattern matching for permutations|journal=[[Information Processing Letters]]|date=March 1998|volume=65|issue=5|pages=277–283|doi=10.1016/S0020-0190(97)00209-3}}</ref> However, PPM can be solved in [[linear time]] when ''k'' is a constant. Indeed, Guillemot and Marx<ref>{{cite journal|last=Guillemot|first=Sylvain|author2=Marx, Daniel|title=Finding small patterns in permutations in linear time|journal=Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms|year=2014|page=20|doi=10.1137/1.9781611973402.7|arxiv=1307.3073|isbn=978-1-61197-338-9|s2cid=1846959}}</ref> showed that PPM can be solved in time <math>2^{O(k^2\log k)} \cdot n</math>, meaning that it is [[fixed-parameter tractable]] with respect to <math>k</math>.

There are several variants on the PPM problem, as surveyed by Bruner and Lackner.<ref>{{citation|title=The computational landscape of permutation patterns|first1=Marie-Louise|last1=Bruner|first2=Martin|last2=Lackner|year=2013|journal=Pure Mathematics and Applications|volume=24|issue=2|pages=83–101|arxiv=1301.0340}}</ref> For example, if the match is required to consist of contiguous entries then the problem can be solved in polynomial time.<ref>{{citation|last1=Kubica|first1=M.|last2=Kulczyński|first2=T.|last3=Radoszewski|first3=J.|last4=Rytter|first4=W.|last5=Waleń|first5=T.|title=A linear time algorithm for consecutive permutation pattern matching|journal=[[Information Processing Letters]]|year=2013|volume=113|issue=12|pages=430–433|doi=10.1016/j.ipl.2013.03.015}}</ref> A different natural variant is obtained when the pattern is restricted to a proper permutation class <math>\mathcal{C}</math>. This problem is known as <math>\mathcal{C}</math>-Pattern PPM and it was shown to be polynomial-time solvable for [[separable permutations]].<ref name="BBL 1998"/> Later, Jelínek and Kynčl<ref name="JK 2017">{{cite conference|title=Hardness of Permutation Pattern Matching|first1=Vít|last1=Jelínek|first2=Jan|last2=Kynčl|year=2017|book-title=Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19|publisher=SIAM|pages=378–396|arxiv=1608.00529|doi=10.1137/1.9781611974782.24}}</ref> completely resolved the complexity of <math>\mbox{Av}(\sigma)</math>-Pattern PPM by showing that it is polynomial-time solvable when <math>\sigma</math> is equal to one of 1, 12, 21, 132, 231, 312 or 213 and NP-complete otherwise.

Another variant is when both the pattern and text are restricted to a proper permutation class <math>\mathcal{C}</math>, in which case the problem is called <math>\mathcal{C}</math>-PPM. For example, Guillemot and Vialette<ref>{{citation|last1=Guillemot|first1=Sylvain|last2=Vialette|first2=Stéphane|contribution=Pattern matching for 321-avoiding permutations|title=Algorithms and Computation|year=2009|volume=5878|series=Lecture Notes in Computer Science|pages=1064–1073|doi=10.1007/978-3-642-10631-6_107|arxiv=1511.01770|isbn=978-3-642-10630-9 }}</ref> showed that <math>\mbox{Av}(321)</math>-PPM could be solved in <math>O(k^2n^6)</math> time. [[Michael H. Albert|Albert]], Lackner, Lackner, and Vatter<ref>{{citation|last1=Albert|first1=Michael | author1-link=Michael H. Albert |last2=Lackner|first2=Marie-Louise|last3=Lackner|first3=Martin|last4=Vatter|first4=Vincent|year=2016|volume=18|issue=2|journal=Discrete Mathematics & Theoretical Computer Science|title=The complexity of pattern matching for 321-avoiding and skew-merged permutations|doi=10.46298/dmtcs.1308 |arxiv=1510.06051|s2cid=5827603 }}</ref> later lowered this to <math>O(kn)</math> and showed that the same bound holds for the class of [[skew-merged permutation]]s. They further asked if the <math>\mathcal{C}</math>-PPM problem can be solved in polynomial time for every fixed proper permutation class <math>\mathcal{C}</math>. This question was answered negatively by Jelínek and Kynčl who showed that <math>\mbox{Av}(4321)</math>-PPM is in fact NP-complete.<ref name="JK 2017" /> Later, Jelínek, Opler, and Pekárek<ref>{{cite conference|title=Griddings of Permutations and Hardness of Pattern Matching|first1=Vít|last1=Jelínek|first2=Michal|last2=Opler|first3=Jakub|last3=Pekárek|year=2021|book-title=46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, August 23-27, 2021, Tallinn, Estonia|publisher=Schloss Dagstuhl - Leibniz-Zentrum für Informatik|issue=2|pages=65:1–65:22|arxiv=2107.10897|doi=10.4230/LIPIcs.MFCS.2021.65|doi-access=free }}</ref> showed that <math>\mbox{Av}(\sigma)</math>-PPM is NP-complete for any <math>\sigma</math> of length at least 4 not symmetric to one of 3412, 3142, 4213, 4123 or 41352.

== Packing densities ==
The permutation π is said to be β-''optimal'' if no permutation of the same length as π has more copies of β.  In his address to the SIAM meeting on Discrete Mathematics in 1992, Wilf defined the ''packing density'' of the permutation β of length ''k'' as

: <math>
\lim_{n\rightarrow\infty}
\frac{\text{number of copies of }\beta\text{ in a }\beta\text{-optimal permutation of length }n}{\displaystyle{n\choose k}}.
</math>

An unpublished argument of [[Fred Galvin]] shows that the quantity inside this [[limit of a sequence|limit]] is nonincreasing for ''n'' &ge; ''k'', and so the limit exists.  When &beta; is monotone, its packing density is clearly 1, and packing densities are invariant under the group of symmetries generated by inverse and reverse, so for permutations of length three, there is only one nontrivial packing density.  Walter Stromquist (unpublished) settled this case by showing that the packing density of 132 is {{math|2{{radic|3}}&nbsp;&minus;&nbsp;3}}, approximately 0.46410.

For permutations β of length four, there are (due to symmetries) seven cases to consider:

{| class="wikitable" style="text-align:center;" border="1"
|-
! β !! packing density !! reference
|-
| &nbsp;1234&nbsp; || 1 || trivial
|-
| &nbsp;1432&nbsp; || root of {{math|''x''<sup>3</sup> &minus; 12''x''<sup>2</sup> + 156''x'' &minus; 64 &cong; 0.42357}} || {{harvtxt|Price|1997}}<ref name="price97">{{citation | last=Price | first=Alkes | title=Packing densities of layered patterns | publisher=University of Pennsylvania | series = Ph.D. thesis | year=1997 | url=https://www.proquest.com/docview/304421853|id={{ProQuest|304421853}} }}.</ref>
|-
| &nbsp;2143&nbsp; || {{math|1={{sfrac|3|8}} = 0.375}} || {{harvtxt|Price|1997}}<ref name="price97"/>
|-
| &nbsp;1243&nbsp; || {{math|1={{sfrac|3|8}} = 0.375}} || {{harvtxt|Albert|Atkinson|Handley|Holton|2002}}<ref>{{Citation
| last1=Albert | first1=Michael H. | author1-link=Michael H. Albert 
| last2=Atkinson | first2=M. D.
| last3=Handley | first3=C. C. 
| last4=Holton | first4=D. A.
| last5=Stromquist | first5=W.
| title=On packing densities of permutations
| year=2002
| journal=[[Electronic Journal of Combinatorics]]
| volume=9
| pages=Article R5, 20 pp
| doi=10.37236/1622 | mr = 1887086
| url=http://www.combinatorics.org/Volume_9/Abstracts/v9i1r5.html| doi-access=free}}.</ref>
|-
| &nbsp;1324&nbsp; || conjectured to be {{math|&cong; 0.244}} || 
|-
| &nbsp;1342&nbsp; || conjectured to be {{math|&cong; 0.19658}} || 
|-
| &nbsp;2413&nbsp; || conjectured to be {{math|&cong; 0.10474}} || 
|}

For the three unknown permutations, there are bounds and conjectures.  {{harvtxt|Price|1997}} used an approximation algorithm which suggests that the packing density of 1324 is around 0.244.<ref name="price97"/>  Birzhan Batkeyev (unpublished) constructed a family of permutations showing that the packing density of 1342 is at least the product of the packing densities of 132 and 1432, approximately 0.19658.  This is conjectured to be the precise packing density of 1342.  {{harvtxt|Presutti|Stromquist|2010}} provided a lower bound on the packing density of 2413.  This lower bound, which can be expressed in terms of an integral, is approximately 0.10474, and conjectured to be the true packing density.<ref>{{ Citation
 | last1=Presutti
 | first1=Cathleen Battiste
 | last2=Stromquist
 | first2=Walter
 | editor1-last=Linton
 | editor1-first=Steve
 | editor2-last=Ruškuc
 | editor2-first=Nik
 | editor3-last=Vatter
 | editor3-first=Vincent
 | title=Permutation Patterns
 | year=2010
 | contribution=Packing rates of measures and a conjecture for the packing density of 2413
 | contribution-url=https://works.swarthmore.edu/fac-math-stat/230
 | publisher=Cambridge University Press
 | series=London Math. Soc. Lecture Notes
 | volume=376
 | pages=287–316
 | doi = 10.1017/CBO9780511902499.015
 | isbn=978-0-521-72834-8
 }}.</ref>

== Superpatterns ==
{{main article|Superpattern}}
A ''k''-'''superpattern''' is a permutation that contains all permutations of length ''k''. For example, 25314 is a 3-superpattern because it contains all 6 permutations of length 3. It is known that ''k''-superpatterns must have length at least ''k''<sup>2</sup>/''e''<sup>2</sup>, where ''e''&nbsp;&asymp; 2.71828 is [[e (mathematical constant)|Euler's number]],<ref>{{citation
 | last = Arratia | first = Richard | authorlink = Richard Arratia
 | journal = [[Electronic Journal of Combinatorics]]
 | mr = 1710623
 | page = Article N1, 4 pp
 | 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> and that there exist ''k''-superpatterns of length ⌈(''k''<sup>2</sup>&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>
This upper bound is conjectured to be best possible, up to lower-order terms.<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>

==Generalizations==
The type of pattern defined above, in which entries do not need to occur consecutively, is called a ''classical'' (permutation) pattern.
If the entries are required to be consecutive, then the pattern is called a ''consecutive pattern''.

There are several ways in which the notion of "pattern" has been generalized.  For example, a ''vincular pattern'' is a permutation containing dashes indicating which adjacent pairs of entries need not occur consecutively.  For example, the permutation 314265 has two copies of the dashed pattern 2&minus;31&minus;4, given by the entries 3426 and 3425.  For a dashed pattern β and any permutation π, we write β(π) for the number of copies of β in π.  Thus the number of inversions in π is 2&minus;1(π), while the number of descents is 21(π).  Going further, the number of ''valleys'' in π is 213(π) + 312(π), while the number of ''peaks'' is 231(π) + 132(π).  These patterns were introduced by {{harvtxt|Babson|Steingrímsson|2000}}, who showed that almost all known [[Mahonian statistic]]s could be expressed in terms of vincular permutations.<ref>{{citation
| last1=Babson | first1=Erik
| last2= Steingrímsson | first2=Einar
| title=Generalized permutation patterns and a classification of the Mahonian statistics
| year=2000
| journal=[[Séminaire Lotharingien de Combinatoire]]
| volume=44
| pages=Research article B44b, 18 pp
| mr = 1758852
| url=http://www.emis.de/journals/SLC/wpapers/s44stein.html}}.</ref>  For example, the [[Major index]] of π is equal to 1&minus;32(π) + 2&minus;31(π) + 3&minus;21(π) + 21(π).

Another generalization is that of a ''barred pattern'', in which some of the entries are barred.  For π to avoid the barred pattern β means that every set of entries of π which form a copy of the nonbarred entries of β can be extended to form a copy of all entries of β.  {{harvtxt|West|1993}} introduced these types of patterns in his study of permutations which could be sorted by passing them twice through a stack.<ref>{{Citation | last1=West | first1=Julian | title=Sorting twice through a stack| mr = 1235186| year=1993 | journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]]| volume=117 | issue=1–2 | pages=303–313| doi = 10.1016/0304-3975(93)90321-J| doi-access=free}}.</ref>  (Note that West's definition of sorting twice through a stack is not the same as sorting with two stacks in series.)  Another example of barred patterns occurs in the work of {{harvtxt|Bousquet-Mélou|Butler|2007}}, who showed that the [[Schubert variety]] corresponding to π is [[Schubert variety#Locally factorial|locally factorial]] if and only if π avoids 1324 and 21<span style="text-decoration: overline;">3</span>54.<ref>{{Citation
| last1=Bousquet-Mélou | first1=Mireille | author1-link = Mireille Bousquet-Mélou
| last2=Butler | first2=Steve |authorlink2=Steve Butler (mathematician)
| title=Forest-like permutations
| year=2007
| journal=[[Annals of Combinatorics]]
| volume=11
| issue=3–4 | pages=335–354
| mr = 2376109
| doi=10.1007/s00026-007-0322-1| arxiv=math/0603617| s2cid=31236417 }}.</ref>

== References ==
{{Reflist|30em}}

==External links==
{{Commonscat|Permutation patterns}}
A conference on permutation patterns has been [https://permutationpatterns.com/history/ held annually since 2003]:
# [http://www.cs.otago.ac.nz/conferences/PP2003/ Permutation Patterns 2003], February 10–14, 2003, [[University of Otago]], Dunedin, New Zealand.
# [https://web.archive.org/web/20041102100037/http://www.mala.ca/math/pp Permutation Patterns 2004], July 5–9, 2004, [[Malaspina University-College]], Nanaimo, British Columbia, Canada.
# [http://math.haifa.ac.il/toufik/conf_2005/pp05.html Permutation Patterns 2005], March 7–11, 2005, [[University of Florida]], Gainesville, Florida, USA.
# [http://www.cs.otago.ac.nz/staffpriv/mike/PP2006/Home.html Permutation Patterns 2006], June 12–16, 2006, [[Reykjavík University]], Reykjavík, Iceland.
# [http://turnbull.mcs.st-and.ac.uk/~pp2007/ Permutation Patterns 2007], June 11–15, 2007, [[University of St. Andrews]], St. Andrews, Scotland.
# [http://www.cs.otago.ac.nz/staffpriv/mike/PP2008/ Permutation Patterns 2008], June 16–20, 2008, [[University of Otago]], Dunedin, New Zealand.
# [http://www.dsi.unifi.it/~PP2009/ Permutation Patterns 2009], July 13–17, 2009, [[Università di Firenze]], Florence, Italy.
# [http://math.dartmouth.edu/~pp2010/ Permutation Patterns 2010], August 9–13, 2010, [[Dartmouth College]], Hanover, New Hampshire, USA.
# [https://web.archive.org/web/20110820182310/http://math.calpoly.edu/PP2011 Permutation Patterns 2011], June 20–24, 2011, [[California Polytechnic State University]], San Luis Obispo, California, USA.
# [http://combinatorics.cis.strath.ac.uk/pp2012/ Permutation Patterns 2012], June 11–15, 2012, [[University of Strathclyde]], Glasgow, Scotland.
# [http://www.lix.polytechnique.fr/pp2013/ Permutation Patterns 2013], July 1–5, 2013, [[Université Paris Diderot]], Paris, France.
# [http://www.etsu.edu/cas/math/pp2014/ Permutation Patterns 2014], July 7–11, 2014, [[East Tennessee State University]], Johnson City, Tennessee, USA.
# [https://sites.google.com/site/pp2015london/ Permutation Patterns 2015], June 15–19, 2015, [http://demorganhouse.org.uk/ De Morgan House], London, England.
# [https://permutationpatterns2016.wordpress.com/ Permutation Patterns 2016], June 27–July 1, 2016, [[Howard University]], Washington, DC, USA.
# [https://pp2017.github.io/ Permutation Patterns 2017], June 26–30, 2017, [[Reykjavík University]], Reykjavík, Iceland.
# [http://2018.permutationpatterns.com/ Permutation Patterns 2018], July 9–13, 2018, [[Dartmouth College]], Hanover, New Hampshire, USA.
# [https://sites.google.com/view/permutation-patterns-2019/ Permutation Patterns 2019], June 17–21, 2019, [[Universität Zürich]], Zürich, Switzerland.
# [https://permutationpatterns.com/2020-virtual-workshop/ Permutation Patterns 2020 Virtual Workshop], June 30–July 1, 2020, hosted by [[Valparaiso University]], Valparaiso, Indiana, USA.
# [http://combinatorics.cis.strath.ac.uk/pp2021/ Permutation Patterns 2021 Virtual Workshop], June 15–16, 2021, hosted by [[University of Strathclyde]], Glasgow, Scotland.
# [https://www.valpo.edu/mathematics-statistics/permutation-patterns-2022/ Permutation Patterns 2022], June 20-24, 2022, [[Valparaiso University]], Valparaiso, Indiana, USA.
# [https://2023.permutationpatterns.com/ Permutation Patterns 2023], July 3-7, 2023, [[University of Burgundy]], Dijon, France.

[[American Mathematical Society]] Special Sessions on Patterns in Permutations have been held at the following meetings:
* [https://www.ams.org/meetings/sectional/2198_program_ss19.html Fall Eastern Sectional Meeting], September 22–23, 2012, [[Rochester Institute of Technology]], Rochester, NY.
* [http://jointmathematicsmeetings.org/meetings/national/jmm2013/2141_program_ss35.html Joint Mathematics Meetings], January 12, 2013, San Diego, CA.
* [https://www.ams.org/meetings/sectional/2212_program_ss15.html Central Fall Sectional Meeting], September 20–21, 2014, [[University of Wisconsin-Eau Claire]], Eau Claire, WI.
* [https://www.ams.org/meetings/sectional/2225_program_ss30.html Spring Eastern Sectional Meeting], March 7–8, 2015, [[Georgetown University]], Washington, DC.

Other permutation patterns meetings:
* [http://math.haifa.ac.il/toufik/conf_work05/work2004.html Workshop on Permutation Patterns], May 29–June 3, 2005, [[University of Haifa]], Haifa, Israel.
* [https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=16071 Pattern Avoidance and Genome Sorting], February 14-19, 2016, [[Schloss Dagstuhl]], Wadern, Germany.
* [https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=18451 Genomics, Pattern Avoidance, and Statistical Mechanics], November 4-9, 2018, [[Schloss Dagstuhl]], Wadern, Germany.
* [https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=23121 Pattern Avoidance, Statistical Mechanics and Computational Complexity], March 19-24, 2023, [[Schloss Dagstuhl]], Wadern, Germany.

Other links:
* [http://www.cs.otago.ac.nz/PermLab PermLab: software for permutation patterns], maintained by [[Michael H. Albert|Michael Albert]].
* [http://math.depaul.edu/~bridget/patterns.html Database of Permutation Pattern Avoidance], maintained by [[Bridget Tenner]].
* [https://permpal.com/ PermPAL: The Permutation Pattern Avoidance Library], a database of algorithmically-derived theorems about permutation classes, maintained by Christian Bean, Émile Nadeau, Jay Pantone and Henning Ulfarsson.

[[Category:Permutation patterns| ]]