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| The '''Stanley–Wilf conjecture''', formulated independently by [[Richard P. Stanley]] and [[Herbert Wilf]] in the late 1980s, states that every [[permutation pattern]] defines a set of permutations whose growth rate is [[Exponential growth|singly exponential]]. It was proved by {{harvs|first1=Adam|last1=Marcus|author1-link=Adam Marcus (mathematician)|first2=Gábor|last2=Tardos|author2-link=Gábor Tardos|year=2004|txt}} and is no longer a conjecture. Marcus and Tardos actually proved a different conjecture, due to {{harvs|first1=Zoltán|last1=Füredi|author1-link=Zoltán Füredi|first2=Péter|last2=Hajnal|year=1992|txt}}, which had been shown to imply the Stanley–Wilf conjecture by {{harvtxt|Klazar|2000}}. | | The name of the author is Figures. I am a meter reader. His spouse doesn't like it the way he does but what he really likes performing is to do aerobics and he's been doing it for fairly a while. Minnesota has always been his house but his spouse wants them to move.<br><br>Also visit my blog post; [http://www.gaysphere.net/user/KJGI http://www.gaysphere.net/user/KJGI] |
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| ==Statement==
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| The Stanley–Wilf conjecture states that for every permutation ''β'', there is a constant ''C'' such that the number |''S''<sub>''n''</sub>(''β'')| of permutations of length ''n'' which avoid ''β'' as a [[permutation pattern]] is at most ''C''<sup>''n''</sup>. As {{harvtxt|Arratia|1999}} observed, this is equivalent to the convergence of the [[Limit (mathematics) |limit]]
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| :<math>\lim_{n\to\infty} |S_n(\beta)|^{1/n}.</math>
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| The upper bound given by Marcus and Tardos for ''C'' is [[Exponential function|exponential]] in the length of ''β''. A stronger conjecture of {{harvtxt|Arratia|1999}} had stated that one could take ''C'' to be {{Nowrap|(''k'' − 1)<sup>2</sup>}}, where ''k'' denotes the length of ''β'', but this conjecture was disproved for the permutation {{Nowrap|1=''β'' = 4231}} by {{harvtxt|Albert|Elder|Rechnitzer|Westcott|2006}}. Indeed, {{harvtxt|Fox|preprint}} has shown that ''C'' is, in fact, exponential in ''k'' for [[almost all]] permutations.
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| ==Allowable growth rates==
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| Not every growth rate of the form ''C''<sup>''n''</sup> may be achieved by a permutation class, regardless of whether it is defined by a single forbidden permutation pattern or a set of forbidden patterns. If the number of permutations in a permutation class grows at more than a polynomial rate, it must grow at least as quickly as the [[Fibonacci number]]s. More specifically, define the growth constant (or Stanley–Wilf limit) of a permutation class ''P'', with ''f''<sub>''P''</sub>(''n'') permutations of length ''n'', to be
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| :<math>\limsup_{n\to\infty} f_P(n)^{1/n}.</math>
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| If the growth constant is zero, then ''f''<sub>''P''</sub>(''n'') must be a polynomial. If it is not zero, then it must be the largest root of a polynomial of the form
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| :<math>1+x+x^2+x^3+\cdots x^{k-1}=x^k,</math>
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| for an integer ''k'' ≥ 2. | |
| For ''k'' = 2, ''C'' is the [[golden ratio]], the base of the growth rate of the Fibonacci numbers. In general, as ''k'' grows larger, these roots approach 2. Thus, in this range, there are only a countably infinite number of growth rates possible.<ref>{{harvtxt|Klazar|2010}}; {{harvtxt|Kaiser|Klazar|2003}}.</ref> However, for every ''C'' > 2.48188 there exists a permutation class (possibly with infinitely many forbidden patterns) whose growth constant is ''C''.<ref>{{harvtxt|Klazar|2010}}; {{harvtxt|Vatter|2010}}.</ref>
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| ==See also==
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| *[[Enumerations of specific permutation classes]] for the growth rates of specific sets defined by permutation patterns
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{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 | issue=2 | pages=96–105 | doi=10.1016/j.aam.2005.05.007}}.
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| *{{Citation | last1=Arratia | first1=Richard | authorlink=Richard Arratia | title=On the Stanley–Wilf conjecture for the number of permutations avoiding a given pattern | mr = 1710623 | year=1999 | journal=[[Electronic Journal of Combinatorics]] | volume=6 | page = N1 | url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v6i1n1}}.
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| *{{Citation | last1=Fox | first1=Jacob | year=preprint | title=Stanley-Wilf limits are typically exponential | id = {{arxiv | id = 1310.8378}}
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| }}.
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| *{{Citation | last1=Füredi | first1=Zoltán | author1-link=Zoltán Füredi | last2=Hajnal | first2=Péter | title=Davenport–Schinzel theory of matrices | mr = 1171777 | year=1992 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]] | volume=103 | issue=3 | pages=233–251 | doi=10.1016/0012-365X(92)90316-8}}.
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| *{{citation
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| | last1 = Kaiser | first1 = Tomáš
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| | last2 = Klazar | first2 = Martin
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| | date = March 2002
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| | issue = 2
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| | journal = Electronic Journal of Combinatorics
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| | mr = 2028280
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| | page = Research paper 10, 20
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| | title = On growth rates of closed permutation classes
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| | url = http://www.combinatorics.org/Volume_9/Abstracts/v9i2r10.html
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| | volume = 9}}.
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| *{{Citation | last1=Klazar | first1=Martin | contribution=The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture | mr = 1798218 | year=2000 | title=Formal Power Series and Algebraic Combinatorics (Moscow, 2000) | publisher = Springer | pages=250–255}}.
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| *{{citation
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| | last = Klazar | first = Martin
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| | contribution = Some general results in combinatorial enumeration
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| | doi = 10.1017/CBO9780511902499.002
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| | location = Cambridge
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| | mr = 2732822
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| | pages = 3–40
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| | publisher = Cambridge Univ. Press
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| | series = London Math. Soc. Lecture Note Ser.
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| | title = Permutation patterns
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| | volume = 376
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| | year = 2010}}.
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| *{{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}}.
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| *{{citation
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| | last = Vatter | first = Vincent
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| | doi = 10.1112/S0025579309000503
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| | issue = 1
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| | journal = Mathematika
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| | mr = 2604993
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| | pages = 182–192
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| | title = Permutation classes of every growth rate above 2.48188
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| | volume = 56
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| | year = 2010}}.
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| ==External links==
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| * [http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/paramath.pdf A Description of The Stanley–Wilf Conjecture] – by [[Doron Zeilberger]].
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| * {{mathworld|urlname=Stanley-WilfConjecture|title=Stanley-Wilf conjecture}}
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| {{DEFAULTSORT:Stanley-Wilf conjecture}}
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| [[Category:Enumerative combinatorics]]
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| [[Category:Theorems in discrete mathematics]]
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| [[Category:Permutation patterns]]
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The name of the author is Figures. I am a meter reader. His spouse doesn't like it the way he does but what he really likes performing is to do aerobics and he's been doing it for fairly a while. Minnesota has always been his house but his spouse wants them to move.
Also visit my blog post; http://www.gaysphere.net/user/KJGI