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| In the study of [[permutation pattern]]s, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements.
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| == Classes avoiding one pattern of length 3 ==
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|
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| There are two symmetry classes and a single Wilf class for single permutations of length three.
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| {| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
| |
| |-
| |
| ! β !! sequence enumerating ''Av<sub>n</sub>''(β) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 123 <br />
| |
| [[stack-sortable permutation|231]]
| |
| || 1, 2, 5, 14, 42, 132, 429, 1430, ... || {{OEIS link|id=A000108}} || algebraic (nonrational) [[Generating function|g.f.]] <br /> [[Catalan numbers]] || {{harvtxt|MacMahon|1915/16}} <br /> {{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 ''Av<sub>n</sub>''(β) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 1342 <br />
| |
| 2413
| |
| || 1, 2, 6, 23, 103, 512, 2740, 15485, ... || {{OEIS link|id=A022558}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bóna|1997}}
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 1234 <br />
| |
| 1243 <br />
| |
| 1432 <br />
| |
| [[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 was given by {{harvtxt|Johansson|Nakamura|preprint}} using functional equations.
| |
| {{harvtxt|Albert|Elder|Rechnitzer|Westcott|2006}} have provided a lower bound and {{harvtxt|Bóna|preprint}} an upper bound for the growth of this class.
| |
| | |
| == 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 ''Av<sub>n</sub>''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence
| |
| |-
| |
| | 123, 321 || 1, 2, 4, 4, 0, 0, 0, 0, ... || n/a || finite
| |
| |-
| |
| | 123, 231 || 1, 2, 4, 7, 11, 16, 22, 29, ... || {{OEIS link|id=A000124}} || polynomial, <math>{n\choose 2}+1</math>
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 123, 132 <br />
| |
| [[Gilbreath shuffle|132, 312]] <br />
| |
| 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 ''Av<sub>n</sub>''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence
| |
| |-
| |
| | 321, 1234 || 1, 2, 5, 13, 25, 25, 0, 0, ... || n/a || finite
| |
| |-
| |
| | 321, 2134 || 1, 2, 5, 13, 30, 61, 112, 190, ... || {{OEIS link|id=A116699}} || polynomial
| |
| |-
| |
| | 132, 4321 || 1, 2, 5, 13, 31, 66, 127, 225, ...|| {{OEIS link|id=A116701}} || polynomial
| |
| |-
| |
| | 321, 1324 || 1, 2, 5, 13, 32, 72, 148, 281, ... || {{OEIS link|id=A179257}} || polynomial
| |
| |-
| |
| | 321, 1342 || 1, 2, 5, 13, 32, 74, 163, 347, ... || {{OEIS link|id=A116702}} || rational [[Generating function|g.f.]]
| |
| |-
| |
| | 321, 2143 || 1, 2, 5, 13, 33, 80, 185, 411, ... || {{OEIS link|id=A088921}} || rational [[Generating function|g.f.]]
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 132, 4312 <br />
| |
| 132, 4231
| |
| || 1, 2, 5, 13, 33, 81, 193, 449, ... || {{OEIS link|id=A005183}} || rational [[Generating function|g.f.]]
| |
| |-
| |
| | 132, 3214 || 1, 2, 5, 13, 33, 82, 202, 497, ... || {{OEIS link|id=A116703}} || rational [[Generating function|g.f.]]
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 321, 2341 <br />
| |
| 321, 3412 <br />
| |
| 321, 3142 <br />
| |
| 132, 1234 <br />
| |
| 132, 4213 <br />
| |
| 132, 4123 <br />
| |
| 132, 3124 <br />
| |
| 132, 2134 <br />
| |
| 132, 3412
| |
| || 1, 2, 5, 13, 34, 89, 233, 610, ... || {{OEIS link|id=A001519}} || rational [[Generating function|g.f.]], <br /> alternate [[Fibonacci number]]s
| |
| |}
| |
| | |
| == Classes avoiding two patterns of length 4 ==
| |
| There are 56 symmetry classes and 38 Wilf equivalence classes, of which 27 have been enumerated.
| |
| | |
| {| class="wikitable" style="text-align:left;" border="1" cell-padding="2"
| |
| |-
| |
| ! B !! sequence enumerating ''Av<sub>n</sub>''(B) !! [[On-Line Encyclopedia of Integer Sequences|OEIS]] !! type of sequence !! exact enumeration reference
| |
| |-
| |
| | 4321, 1234 || 1, 2, 6, 22, 86, 306, 882, 1764, ... || {{OEIS link|id=A206736 }} || finite || [[Erdős–Szekeres theorem]]
| |
| |-
| |
| | 4312, 1234 || 1, 2, 6, 22, 86, 321, 1085, 3266, ... || {{OEIS link|id=A116705}} || polynomial || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4321, 3124 || 1, 2, 6, 22, 86, 330, 1198, 4087, ... || {{OEIS link|id=A116708}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4312, 2134 || 1, 2, 6, 22, 86, 330, 1206, 4174, ... || {{OEIS link|id=A116706}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4321, 1324 || 1, 2, 6, 22, 86, 332, 1217, 4140, ... || {{OEIS link|id=A165524}} || polynomial || {{harvtxt|Vatter|2012}}
| |
| |-
| |
| | 4321, 2143 || 1, 2, 6, 22, 86, 333, 1235, 4339, ... || {{OEIS link|id=A165525}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
| |
| |-
| |
| | 4312, 1324 || 1, 2, 6, 22, 86, 335, 1266, 4598, ... || {{OEIS link|id=A165526}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
| |
| |-
| |
| | 4231, 2143 || 1, 2, 6, 22, 86, 335, 1271, 4680, ... || {{OEIS link|id=A165527}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2011}}
| |
| |-
| |
| | 4231, 1324 || 1, 2, 6, 22, 86, 336, 1282, 4758, ... || {{OEIS link|id=A165528}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|2009}}
| |
| |-
| |
| | 4213, 2341 || 1, 2, 6, 22, 86, 336, 1290, 4870, ... || {{OEIS link|id=A116709}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4312, 2143 || 1, 2, 6, 22, 86, 337, 1295, 4854, ... || {{OEIS link|id=A165529}} || rational [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Brignall|2012}}
| |
| |-
| |
| | 4213, 1243 || 1, 2, 6, 22, 86, 337, 1299, 4910, ... || {{OEIS link|id=A116710}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4321, 3142 || 1, 2, 6, 22, 86, 338, 1314, 5046, ... || {{OEIS link|id=A165530}} || rational [[Generating function|g.f.]] || {{harvtxt|Vatter|2012}}
| |
| |-
| |
| | 4213, 1342 || 1, 2, 6, 22, 86, 338, 1318, 5106, ... || {{OEIS link|id=A116707}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 4312, 2341 || 1, 2, 6, 22, 86, 338, 1318, 5110, ... || {{OEIS link|id=A116704}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 3412, 2143 || 1, 2, 6, 22, 86, 340, 1340, 5254, ... || {{OEIS link|id=A029759}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Atkinson|1998}}
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 4321, 4123<br />
| |
| 4321, 3412<br />
| |
| 4123, 3142<br />
| |
| 4123, 2143
| |
| || 1, 2, 6, 22, 86, 342, 1366, 5462, ... || {{OEIS link|id=A047849}} || rational [[Generating function|g.f.]] || {{harvtxt|Kremer|Shiu|2003}}
| |
| |-
| |
| | 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}}
| |
| |-
| |
| | 4231, 3214 || 1, 2, 6, 22, 87, 352, 1428, 5768, ... || {{OEIS link|id=A165532}} || ||
| |
| |-
| |
| | 4213, 1432 || 1, 2, 6, 22, 87, 352, 1434, 5861, ... || {{OEIS link|id=A165533}} || ||
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 4312, 3421<br />
| |
| 4213, 2431
| |
| || 1, 2, 6, 22, 87, 354, 1459, 6056, ... || {{OEIS link|id=A164651}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Callan|preprint1}} determined the enumeration;<br />{{harvtxt|Le|2005}} established the Wilf-equivalence.
| |
| |-
| |
| | 4312, 3124 || 1, 2, 6, 22, 88, 363, 1507, 6241, ... || {{OEIS link|id=A165534}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Pantone|preprint}}
| |
| |-
| |
| | 4231, 3124 || 1, 2, 6, 22, 88, 363, 1508, 6255, ... || {{OEIS link|id=A165535}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|preprint}}
| |
| |-
| |
| | 4312, 3214 || 1, 2, 6, 22, 88, 365, 1540, 6568, ... || {{OEIS link|id=A165536}} || ||
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 4231, 3412<br />
| |
| 4231, 3142<br />
| |
| 4213, 3241<br />
| |
| 4213, 3124<br />
| |
| 4213, 2314
| |
| || 1, 2, 6, 22, 88, 366, 1552, 6652, ... || {{OEIS link|id=A032351}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Bóna|1998}}
| |
| |-
| |
| | 4213, 2143 || 1, 2, 6, 22, 88, 366, 1556, 6720, ... || {{OEIS link|id=A165537}} || ||
| |
| |-
| |
| | 4312, 3142 || 1, 2, 6, 22, 88, 367, 1568, 6810, ... || {{OEIS link|id=A165538}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|preprint}}
| |
| |-
| |
| | 4213, 3421 || 1, 2, 6, 22, 88, 367, 1571, 6861, ... || {{OEIS link|id=A165539}} || ||
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 4213, 3412<br />
| |
| 4123, 3142
| |
| || 1, 2, 6, 22, 88, 368, 1584, 6968, ... || {{OEIS link|id=A109033}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Le|2005}}
| |
| |-
| |
| | 4321, 3214 || 1, 2, 6, 22, 89, 376, 1611, 6901, ... || {{OEIS link|id=A165540}} || ||
| |
| |-
| |
| | 4213, 3142 || 1, 2, 6, 22, 89, 379, 1664, 7460, ... || {{OEIS link|id=A165541}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Albert|Atkinson|Vatter|preprint}}
| |
| |-
| |
| | 4231, 4123 || 1, 2, 6, 22, 89, 380, 1677, 7566, ... || {{OEIS link|id=A165542}} || ||
| |
| |-
| |
| | 4321, 4213 || 1, 2, 6, 22, 89, 380, 1678, 7584, ... || {{OEIS link|id=A165543}} || algebraic (nonrational) [[Generating function|g.f.]] || {{harvtxt|Callan|preprint2}}
| |
| |-
| |
| | 4123, 3412 || 1, 2, 6, 22, 89, 381, 1696, 7781, ... || {{OEIS link|id=A165544}} || ||
| |
| |-
| |
| | 4312, 4123 || 1, 2, 6, 22, 89, 382, 1711, 7922, ... || {{OEIS link|id=A165545}} || ||
| |
| |-
| |
| | <!-- Wilf equivalence: -->
| |
| 4321, 4312<br />
| |
| 4312, 4231<br />
| |
| 4312, 4213<br />
| |
| 4312, 3412<br />
| |
| 4231, 4213<br />
| |
| 4213, 4132<br />
| |
| 4213, 4123<br />
| |
| 4213, 2413<br />
| |
| 4213, 3214<br />
| |
| [[Separable permutation|3142, 2413]]
| |
| || 1, 2, 6, 22, 90, 394, 1806, 8558, ... || {{OEIS link|id=A006318}} || algebraic (nonrational) [[Generating function|g.f.]]<br />[[Schröder number]]s || {{harvtxt|Kremer|2000}}, {{harvtxt|Kremer|2003}}
| |
| |-
| |
| | 3412, 2413 || 1, 2, 6, 22, 90, 395, 1823, 8741, ... || {{OEIS link|id=A165546}} || ||
| |
| |-
| |
| | 4321, 4231 || 1, 2, 6, 22, 90, 396, 1837, 8864, ... || {{OEIS link|id=A053617}} || ||
| |
| |}
| |
| | |
| == See also ==
| |
| * [[Baxter permutation]]
| |
| * [[Riffle shuffle permutation]]
| |
| | |
| == References ==
| |
| *{{Citation | last1=Albert | first1=Michael H. | author1-link=Michael H. Albert | last2=Elder | first2=Murray | last3=Rechnitzer | first3=Andrew | last4=Westcott | first4=P. | last5=Zabrocki | first5=Mike | title=On the Stanley-Wilf limit of 4231-avoiding permutations and a conjecture of Arratia | mr=2199982 | year=2006 | journal=Advances in Applied Mathematics | volume=36 | pages=96–105 | doi=10.1016/j.aam.2005.05.007}}.
| |
| *{{Citation
| |
| | last1=Albert | first1=Michael H. | author1-link=Michael H. Albert
| |
| | last2=Atkinson | first2=M. D.
| |
| | last3=Brignall | first3=Robert
| |
| | title=The enumeration of permutations avoiding 2143 and 4231
| |
| | year=2011
| |
| | journal=[[Pure Mathematics and Applications]]
| |
| | volume=22
| |
| | pages=87–98
| |
| | url=http://www.mat.unisi.it/newsito/puma/public_html/22_2/albert_atkinson_brignall.pdf}}.
| |
| *{{Citation
| |
| | last1=Albert | first1=Michael H. | author1-link=Michael H. Albert
| |
| | last2=Atkinson | first2=M. D.
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| | last3=Brignall | first3=Robert
| |
| | title=The enumeration of three pattern classes using monotone grid classes
| |
| | year=2012
| |
| | journal=[[Electronic Journal of Combinatorics]]
| |
| | volume=19 (3)
| |
| | pages=P20, 34 pp.
| |
| | url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p20}}.
| |
| *{{Citation
| |
| | last1=Albert | first1=Michael H. | author1-link=Michael H. Albert
| |
| | last2=Atkinson | first2=M. D.
| |
| | last3=Vatter | first3=Vincent
| |
| | title=Counting 1324, 4231-avoiding permutations
| |
| | year=2009
| |
| | journal=[[Electronic Journal of Combinatorics]]
| |
| | volume=16 (1)
| |
| | pages=R136, 9 pp.
| |
| | mr=2577304
| |
| | url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r136}}.
| |
| *{{Citation
| |
| | last1=Albert | first1=Michael H. | author1-link=Michael H. Albert
| |
| | last2=Atkinson | first2=M. D.
| |
| | last3=Vatter | first3=Vincent
| |
| | title=Inflations of geometric grid classes: three case studies
| |
| | year=preprint
| |
| | id = {{arxiv | id = 1209.0425}}
| |
| }}.
| |
| *{{Citation
| |
| | last1=Atkinson | first1=M. D.
| |
| | title=Permutations which are the union of an increasing and a decreasing subsequence
| |
| | year=1998
| |
| | journal=[[Electronic Journal of Combinatorics]]
| |
| | volume=5
| |
| | pages=R6, 13 pp.
| |
| | mr=1490467
| |
| | url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r6}}.
| |
| *{{Citation
| |
| | last1=Atkinson | first1=M. D.
| |
| | title=Restricted permutations
| |
| | year=1999
| |
| | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]
| |
| | volume=195
| |
| | pages=27–38
| |
| | mr=1663866
| |
| | doi=10.1016/S0012-365X(98)00162-9}}.
| |
| *{{Citation
| |
| | last1=Atkinson | first1=M. D.
| |
| | last2=Sagan | first2=Bruce E. | author2-link=Bruce Sagan
| |
| | last3=Vatter | first3=Vincent
| |
| | title=Counting (3+1)-avoiding permutations
| |
| | year=2012
| |
| | journal=[[European Journal of Combinatorics]]
| |
| | volume=33
| |
| | pages=49–61
| |
| | mr=2854630
| |
| | doi=10.1016/j.ejc.2011.06.006}}.
| |
| *{{Citation
| |
| | last1=Bóna | first1=Miklós | authorlink = Miklós Bóna
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| | |
| [[Category:Enumerative combinatorics]]
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| [[Category:Permutation patterns]]
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