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In the study of [[permutation pattern]]s, there has been considerable interest in enumerating specific [[permutation class]]es, especially those with relatively few basis elements. This area of study has turned up unexpected instances of [[Wilf equivalence]], where two seemingly-unrelated permutation classes have the same number of permutations of each length.

== Classes avoiding one pattern of length 3 ==

There are two symmetry classes and a single [[Wilf class]] for single permutations of length three.

{| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
|-
! β !! sequence enumerating ''Avn''(β) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
|-
| 
[https://permpal.com/perms/basis/123/ 123] 
[[stack-sortable permutation|231]]
|| 1, 2, 5, 14, 42, 132, 429, 1430, ... || {{OEIS link|id=A000108}} || algebraic (nonrational) [[Generating function|g.f.]] [[Catalan numbers]] || {{harvtxt|MacMahon|1916}} {{harvtxt|Knuth|1968}}
|}

== Classes avoiding one pattern of length 4 ==

There are seven symmetry classes and three Wilf classes for single permutations of length four.

{| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
|-
! β !! sequence enumerating ''Avn''(β) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
|-
| 
[https://permpal.com/perms/basis/1342/ 1342] 
[https://permpal.com/perms/basis/2413/ 2413]
|| 1, 2, 6, 23, 103, 512, 2740, 15485, ... || {{OEIS link|id=A022558}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bóna|1997}}
|-
| 
[https://permpal.com/perms/basis/1234/ 1234] 
[https://permpal.com/perms/basis/1243/ 1243] 
[https://permpal.com/perms/basis/1432/ 1432] 
[[Vexillary permutation|2143]]
|| 1, 2, 6, 23, 103, 513, 2761, 15767, ... || {{OEIS link|id=A005802}} || [[Holonomic function|holonomic]] (nonalgebraic) [[Generating function|g.f.]] || {{harvtxt|Gessel|1990}}
|-
| 1324 || 1, 2, 6, 23, 103, 513, 2762, 15793, ... || {{OEIS link|id=A061552}} || ||
|}

No non-recursive formula counting 1324-avoiding permutations is known. A recursive formula was given by {{harvtxt|Marinov|Radoičić|2003}}.
A more efficient algorithm using functional equations was given by {{harvtxt|Johansson|Nakamura|2014}}, which was enhanced by {{harvtxt|Conway|Guttmann|2015}}, and then further enhanced by {{harvtxt|Conway|Guttmann|Zinn-Justin|2018}} who give the first 50 terms of the enumeration.
{{harvtxt|Bevan|Brignall|Elvey Price|Pantone|2020}} currently have the best rigorously established lower and upper bounds for the growth rate of this class, having established that this growth rate lies in the interval [10.271, 13.5].

== Classes avoiding two patterns of length 3 ==

There are five symmetry classes and three Wilf classes, all of which were enumerated in {{harvtxt|Simion|Schmidt|1985}}.

{| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
|-
! B !! sequence enumerating ''Avn''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence
|-
| [https://permpal.com/perms/basis/123_321/ 123, 321] || 1, 2, 4, 4, 0, 0, 0, 0, ... || n/a || finite
|-
| [https://permpal.com/perms/basis/213_321/ 213, 321] || 1, 2, 4, 7, 11, 16, 22, 29, ... || {{OEIS link|id=A000124}} || polynomial, <math>{n\choose 2}+1</math>
|-
| 
[https://permpal.com/perms/basis/231_321/ 231, 321] 
[[Gilbreath shuffle|132, 312]] 
[[Layered permutation|231, 312]]
|| 1, 2, 4, 8, 16, 32, 64, 128, ... || {{OEIS link|id=A000079}} || rational [[Generating function|g.f.]], <math>2^{n-1}</math>
|}

== Classes avoiding one pattern of length 3 and one of length 4 ==

There are eighteen symmetry classes and nine Wilf classes, all of which have been enumerated. For these results, see {{harvtxt|Atkinson|1999}} or {{harvtxt|West|1996}}.

{| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
|-
! B !! sequence enumerating ''Avn''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence
|-
| [https://permpal.com/perms/basis/321_1234/ 321, 1234] || 1, 2, 5, 13, 25, 25, 0, 0, ... || n/a || finite
|-
| [https://permpal.com/perms/basis/321_2134/ 321, 2134] || 1, 2, 5, 13, 30, 61, 112, 190, ... || {{OEIS link|id=A116699}} || polynomial
|-
| [https://permpal.com/perms/basis/132_4321/ 132, 4321] || 1, 2, 5, 13, 31, 66, 127, 225, ...|| {{OEIS link|id=A116701}} || polynomial
|-
| [https://permpal.com/perms/basis/321_1324/ 321, 1324] || 1, 2, 5, 13, 32, 72, 148, 281, ... || {{OEIS link|id=A179257}} || polynomial
|-
| [https://permpal.com/perms/basis/321_1342/ 321, 1342] || 1, 2, 5, 13, 32, 74, 163, 347, ... || {{OEIS link|id=A116702}} || rational [[Generating function|g.f.]]
|-
| [https://permpal.com/perms/basis/321_2143/ 321, 2143] || 1, 2, 5, 13, 33, 80, 185, 411, ... || {{OEIS link|id=A088921}} || rational [[Generating function|g.f.]]
|-
| 
[https://permpal.com/perms/basis/132_4312/ 132, 4312] 
[https://permpal.com/perms/basis/132_4231/ 132, 4231]
|| 1, 2, 5, 13, 33, 81, 193, 449, ... || {{OEIS link|id=A005183}} || rational [[Generating function|g.f.]]
|-
| [https://permpal.com/perms/basis/132_3214/ 132, 3214] || 1, 2, 5, 13, 33, 82, 202, 497, ... || {{OEIS link|id=A116703}} || rational [[Generating function|g.f.]]
|-
| 
[https://permpal.com/perms/basis/321_2341/ 321, 2341] 
[https://permpal.com/perms/basis/321_3412/ 321, 3412] 
[https://permpal.com/perms/basis/321_3142/ 321, 3142] 
[https://permpal.com/perms/basis/132_1234/ 132, 1234] 
[https://permpal.com/perms/basis/132_4213/ 132, 4213] 
[https://permpal.com/perms/basis/132_4123/ 132, 4123] 
[https://permpal.com/perms/basis/132_3124/ 132, 3124] 
[https://permpal.com/perms/basis/132_2134/ 132, 2134] 
[https://permpal.com/perms/basis/132_3412/ 132, 3412]
|| 1, 2, 5, 13, 34, 89, 233, 610, ... || {{OEIS link|id=A001519}} || rational [[Generating function|g.f.]], alternate [[Fibonacci number]]s
|}

== Classes avoiding two patterns of length 4 ==
[[File:Heatmaps of all non-finite permutation classes avoiding two length four patterns 11 by 5.png|thumb|right|Heatmaps of classes avoiding two patterns of length 4.]]
There are 56 symmetry classes and 38 Wilf equivalence classes. Only 3 of these remain unenumerated, and their [[generating function]]s are conjectured not to satisfy any [[algebraic differential equation]] (ADE) by {{harvtxt|Albert|Homberger|Pantone|Shar|2018}}; in particular, their conjecture would imply that these generating functions are not [[Holonomic function|D-finite]].

Heatmaps of each of the non-finite classes are shown on the right. The lexicographically minimal symmetry is used for each class, and the classes are ordered in lexicographical order. To create each heatmap, one million permutations of length 300 were sampled uniformly at random from the class. The color of the point <math>(i,j)</math> represents how many permutations have value <math>j</math> at index <math>i</math>. Higher resolution versions can be obtained at [https://permpal.com/ PermPal]

{| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
|-
! B !! sequence enumerating ''Avn''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
|-
| [https://permpal.com/perms/basis/4321_1234/ 4321, 1234] || 1, 2, 6, 22, 86, 306, 882, 1764, ... || {{OEIS link|id=A206736 }} || finite || [[Erdős–Szekeres theorem#Permutation pattern interpretation|Erdős–Szekeres theorem]]
|-
| [https://permpal.com/perms/basis/4312_1234/ 4312, 1234] || 1, 2, 6, 22, 86, 321, 1085, 3266, ... || {{OEIS link|id=A116705}} || polynomial || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4321_3124/ 4321, 3124] || 1, 2, 6, 22, 86, 330, 1198, 4087, ... || {{OEIS link|id=A116708}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4312_2134/ 4312, 2134] || 1, 2, 6, 22, 86, 330, 1206, 4174, ... || {{OEIS link|id=A116706}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4321_1324/ 4321, 1324] || 1, 2, 6, 22, 86, 332, 1217, 4140, ... || {{OEIS link|id=A165524}} || polynomial || {{harvtxt|Vatter|2012}}
|-
| [https://permpal.com/perms/basis/4321_2143/ 4321, 2143] || 1, 2, 6, 22, 86, 333, 1235, 4339, ... || {{OEIS link|id=A165525}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
|-
| [https://permpal.com/perms/basis/4312_1324/ 4312, 1324] || 1, 2, 6, 22, 86, 335, 1266, 4598, ... || {{OEIS link|id=A165526}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
|-
| [https://permpal.com/perms/basis/4231_2143/ 4231, 2143] || 1, 2, 6, 22, 86, 335, 1271, 4680, ... || {{OEIS link|id=A165527}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2011}}
|-
| [https://permpal.com/perms/basis/4231_1324/ 4231, 1324] || 1, 2, 6, 22, 86, 336, 1282, 4758, ... || {{OEIS link|id=A165528}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|2009}}
|-
| [https://permpal.com/perms/basis/4213_2341/ 4213, 2341] || 1, 2, 6, 22, 86, 336, 1290, 4870, ... || {{OEIS link|id=A116709}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4312_2143/ 4312, 2143] || 1, 2, 6, 22, 86, 337, 1295, 4854, ... || {{OEIS link|id=A165529}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
|-
| [https://permpal.com/perms/basis/4213_1243/ 4213, 1243] || 1, 2, 6, 22, 86, 337, 1299, 4910, ... || {{OEIS link|id=A116710}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4321_3142/ 4321, 3142] || 1, 2, 6, 22, 86, 338, 1314, 5046, ... || {{OEIS link|id=A165530}} || rational [[Generating function|g.f.]] || {{harvtxt|Vatter|2012}}
|-
| [https://permpal.com/perms/basis/4213_1342/ 4213, 1342] || 1, 2, 6, 22, 86, 338, 1318, 5106, ... || {{OEIS link|id=A116707}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [https://permpal.com/perms/basis/4312_2341/ 4312, 2341] || 1, 2, 6, 22, 86, 338, 1318, 5110, ... || {{OEIS link|id=A116704}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}; {{harvtxt|Vatter|2006}}
|-
| [[Skew-merged permutation|3412, 2143]] || 1, 2, 6, 22, 86, 340, 1340, 5254, ... || {{OEIS link|id=A029759}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Atkinson|1998}}
|-
| 
[https://permpal.com/perms/basis/4321_4123/ 4321, 4123] 
[https://permpal.com/perms/basis/4321_3412/ 4321, 3412] 
[https://permpal.com/perms/basis/4123_3214/ 4123, 3214] 
[https://permpal.com/perms/basis/4123_2143/ 4123, 2143]
|| 1, 2, 6, 22, 86, 342, 1366, 5462, ... || {{OEIS link|id=A047849}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
|-
| [https://permpal.com/perms/basis/4123_2341/ 4123, 2341] || 1, 2, 6, 22, 87, 348, 1374, 5335, ... || {{OEIS link|id=A165531}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Atkinson|Sagan|Vatter|2012}}
|-
| [https://permpal.com/perms/basis/4231_3214/ 4231, 3214] || 1, 2, 6, 22, 87, 352, 1428, 5768, ... || {{OEIS link|id=A165532}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Miner|2016}}
|-
| [https://permpal.com/perms/basis/4213_1432/ 4213, 1432] || 1, 2, 6, 22, 87, 352, 1434, 5861, ... || {{OEIS link|id=A165533}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Miner|2016}}
|-
| 
[https://permpal.com/perms/basis/4312_3421/ 4312, 3421] 
[https://permpal.com/perms/basis/4213_2431/ 4213, 2431]
|| 1, 2, 6, 22, 87, 354, 1459, 6056, ... || {{OEIS link|id=A164651}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Le|2005}} proved the Wilf-equivalence; {{harvtxt|Callan|2013a}} established the [[Generating function|g.f.]].
|-
| [https://permpal.com/perms/basis/4312_3124/ 4312, 3124] || 1, 2, 6, 22, 88, 363, 1507, 6241, ... || {{OEIS link|id=A165534}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Pantone|2017}}
|-
| [https://permpal.com/perms/basis/4231_3124/ 4231, 3124] || 1, 2, 6, 22, 88, 363, 1508, 6255, ... || {{OEIS link|id=A165535}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|2014}}
|-
| [https://permpal.com/perms/basis/4312_3214/ 4312, 3214] || 1, 2, 6, 22, 88, 365, 1540, 6568, ... || {{OEIS link|id=A165536}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Miner|2016}}
|-
| 
[https://permpal.com/perms/basis/4231_3412/ 4231, 3412] 
[https://permpal.com/perms/basis/4231_3142/ 4231, 3142] 
[https://permpal.com/perms/basis/4213_3241/ 4213, 3241] 
[https://permpal.com/perms/basis/4213_3124/ 4213, 3124] 
[https://permpal.com/perms/basis/4213_2314/ 4213, 2314]
|| 1, 2, 6, 22, 88, 366, 1552, 6652, ... || {{OEIS link|id=A032351}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bóna|1998}}
|-
| [https://permpal.com/perms/basis/4213_2143/ 4213, 2143] || 1, 2, 6, 22, 88, 366, 1556, 6720, ... || {{OEIS link|id=A165537}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bevan|2016b}}
|-
| [https://permpal.com/perms/basis/4312_3142/ 4312, 3142] || 1, 2, 6, 22, 88, 367, 1568, 6810, ... || {{OEIS link|id=A165538}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|2014}}
|-
| [https://permpal.com/perms/basis/4213_3421/ 4213, 3421] || 1, 2, 6, 22, 88, 367, 1571, 6861, ... || {{OEIS link|id=A165539}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bevan|2016a}}
|-
| 
[https://permpal.com/perms/basis/4213_3412/ 4213, 3412] 
[https://permpal.com/perms/basis/4123_3142/ 4123, 3142]
|| 1, 2, 6, 22, 88, 368, 1584, 6968, ... || {{OEIS link|id=A109033}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Le|2005}}
|-
| [https://permpal.com/perms/basis/4321_3214/ 4321, 3214] || 1, 2, 6, 22, 89, 376, 1611, 6901, ... || {{OEIS link|id=A165540}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bevan|2016a}}
|-
| [https://permpal.com/perms/basis/4213_3142/ 4213, 3142] || 1, 2, 6, 22, 89, 379, 1664, 7460, ... || {{OEIS link|id=A165541}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|2014}}
|-
| [https://permpal.com/perms/basis/4231_4123/ 4231, 4123] || 1, 2, 6, 22, 89, 380, 1677, 7566, ... || {{OEIS link|id=A165542}} || conjectured to not satisfy any [[algebraic differential equation|ADE]]; see {{harvtxt|Albert|Homberger|Pantone|Shar|2018}} ||
|-
| [https://permpal.com/perms/basis/4321_4213/ 4321, 4213] || 1, 2, 6, 22, 89, 380, 1678, 7584, ... || {{OEIS link|id=A165543}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Callan|2013b}}; see also {{harvtxt|Bloom|Vatter|2016}}
|-
| [https://permpal.com/perms/basis/4123_3412/ 4123, 3412] || 1, 2, 6, 22, 89, 381, 1696, 7781, ... || {{OEIS link|id=A165544}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Miner|Pantone|2018}}
|-
| [https://permpal.com/perms/basis/4312_4123/ 4312, 4123] || 1, 2, 6, 22, 89, 382, 1711, 7922, ... || {{OEIS link|id=A165545}} || conjectured to not satisfy any [[algebraic differential equation|ADE]]; see {{harvtxt|Albert|Homberger|Pantone|Shar|2018}} ||
|-
| 
[https://permpal.com/perms/basis/4321_4312/ 4321, 4312] 
[https://permpal.com/perms/basis/4312_4231/ 4312, 4231] 
[https://permpal.com/perms/basis/4312_4213/ 4312, 4213] 
[https://permpal.com/perms/basis/4312_3412/ 4312, 3412] 
[https://permpal.com/perms/basis/4231_4213/ 4231, 4213] 
[https://permpal.com/perms/basis/4213_4132/ 4213, 4132] 
[https://permpal.com/perms/basis/4213_4123/ 4213, 4123] 
[https://permpal.com/perms/basis/4213_2413/ 4213, 2413] 
[https://permpal.com/perms/basis/4213_3214/ 4213, 3214] 
[[Separable permutation|3142, 2413]]
|| 1, 2, 6, 22, 90, 394, 1806, 8558, ... || {{OEIS link|id=A006318}} || [[Schröder number]]s algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Kremer|2000}}; see also {{harvtxt|Kremer|2003}}
|-
| [https://permpal.com/perms/basis/3412_2413/ 3412, 2413] || 1, 2, 6, 22, 90, 395, 1823, 8741, ... || {{OEIS link|id=A165546}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Miner|Pantone|2018}}
|-
| [https://permpal.com/perms/basis/4321_4231/ 4321, 4231] || 1, 2, 6, 22, 90, 396, 1837, 8864, ... || {{OEIS link|id=A053617}} || conjectured to not satisfy any [[algebraic differential equation|ADE]]; see {{harvtxt|Albert|Homberger|Pantone|Shar|2018}} ||
|}

== See also ==
* [[Baxter permutation]]
* [[Riffle shuffle permutation]]

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==External links==
The [http://math.depaul.edu/~bridget/patterns.html Database of Permutation Pattern Avoidance], maintained by [[Bridget Tenner]], contains details of the enumeration of many other permutation classes with relatively few basis elements.

[[Category:Enumerative combinatorics]]
[[Category:Permutation patterns]]

Enumerations of specific permutation classes - Revision history

imported>OAbot: Open access bot: arxiv updated in citation with #oabot.