Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It...
Transcript of Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It...
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Nouvelles perspectives sur l’énumération desclasses de permutations
Michael Albert
Department of Computer Science, University of Otago
Sep 7, 2012, LaBRI
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New perspectives on the enumeration ofpermutation classes
Michael Albert
Department of Computer Science, University of Otago
Sep 7, 2012, LaBRI
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Permutation classes
DefinitionA permutation class is a collection of permutations, C, withthe property that, if π ∈ C and we erase some points from itsplot, then the permutation defined by the remaining points isalso in C.
492713685 ∈ C 21543 ∈ Cimplies
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Permutation classes
◮ Permutation classes are simply downward closed sets inthe subpermutation ordering
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Permutation classes
◮ Permutation classes are simply downward closed sets inthe subpermutation ordering
◮ The objective is to try to understand the structure ofpermutation classes (or to identify when this is possible)
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Permutation classes
◮ Permutation classes are simply downward closed sets inthe subpermutation ordering
◮ The objective is to try to understand the structure ofpermutation classes (or to identify when this is possible)
◮ Enumeration is a consequence or symptom of suchunderstanding
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Permutation classes
◮ Permutation classes are simply downward closed sets inthe subpermutation ordering
◮ The objective is to try to understand the structure ofpermutation classes (or to identify when this is possible)
◮ Enumeration is a consequence or symptom of suchunderstanding
◮ If X is a set of permutations, then Av(X ) is the permutationclass consisting of those permutations which do notdominate any permutation of X
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
◮ Av(1234) was enumerated by Gessel (1990)
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
◮ Av(1234) was enumerated by Gessel (1990)◮ Av(1342) by Bóna (1997)
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
◮ Av(1234) was enumerated by Gessel (1990)◮ Av(1342) by Bóna (1997)◮ The enumeration of Av(1324) is still unknown
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
◮ Av(1234) was enumerated by Gessel (1990)◮ Av(1342) by Bóna (1997)◮ The enumeration of Av(1324) is still unknown◮ For single element bases of length 5 or more, nothing
much is known in the non-monotone case
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First steps◮ It seems plausible that the fewer permutations there are in
a class, or alternatively the more restrictive the conditionsof membership, then the more likely it is to have goodstructure
◮ Simion and Schmidt (1985) successfully enumerated allclasses Av(X ) for X ⊆ S3
◮ For a single avoided pattern of length 4, it turns out thereare only three possible enumerations (based on bothspecific and general symmetries)
◮ Av(1234) was enumerated by Gessel (1990)◮ Av(1342) by Bóna (1997)◮ The enumeration of Av(1324) is still unknown◮ For single element bases of length 5 or more, nothing
much is known in the non-monotone case◮ All doubleton bases of lengths 4 and 3, and many of length
4 and 4 are known
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Stanley-Wilf conjecture
Relative to the set of all permutations, proper permutationclasses are small. Specifically:
TheoremLet C be a proper permutation class. Then, the growth rate of C,
gr(C) = lim sup |C ∩ Sn|1/n
is finite.
This was known as the Stanley-Wilf conjecture and it wasproven in 2004 by Marcus and Tardos.The obvious next questions are:
◮ What growth rates can occur?◮ What can be said about classes of particular growth rates?
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Antichains
The subpermutation order contains infinite antichains.
Consequently, there exist 2ℵ0 distinct enumerationsequences for permutation classes – we must be carefulnot to try to do too much.
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Small growth rates◮ Kaiser and Klazar (EJC, 2003) showed that the only
possible values of gr(C) less than 2 are the greatestpositive solutions of:
xk − xk−1 − xk−2 − · · · − x − 1 = 0
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Small growth rates◮ Kaiser and Klazar (EJC, 2003) showed that the only
possible values of gr(C) less than 2 are the greatestpositive solutions of:
xk − xk−1 − xk−2 − · · · − x − 1 = 0
◮ Vatter (Mathematika, 2010) showed that every real numberlarger than the unique real solution (approx. 2.482) of
x5 − 2x4 − 2x2 − 2x − 1 = 0
occurs as a growth rate
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Small growth rates◮ Kaiser and Klazar (EJC, 2003) showed that the only
possible values of gr(C) less than 2 are the greatestpositive solutions of:
xk − xk−1 − xk−2 − · · · − x − 1 = 0
◮ Vatter (Mathematika, 2010) showed that every real numberlarger than the unique real solution (approx. 2.482) of
x5 − 2x4 − 2x2 − 2x − 1 = 0
occurs as a growth rate◮ V (PLMS, 2011) further showed that the smallest growth
rate of a class containing an infinite antichain is the uniquepositive solution, κ ≃ 2.20557 of
x3 − 2x2 − 1 = 0
and completely characterized the set of possible growthrates below κ
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Simple permutations
DefinitionA permutation is simple if it contains no nontrivial consecutivesubsequence whose values are also consecutive (though notnecessarily in order)
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Simple permutations
DefinitionA permutation is simple if it contains no nontrivial consecutivesubsequence whose values are also consecutive (though notnecessarily in order)
2 5 7 4 6 1 3
Not simple
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Simple permutations
DefinitionA permutation is simple if it contains no nontrivial consecutivesubsequence whose values are also consecutive (though notnecessarily in order)
2 5 7 3 6 1 4
Simple
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Simple permutations
DefinitionA permutation is simple if it contains no nontrivial consecutivesubsequence whose values are also consecutive (though notnecessarily in order)
◮ Simple permutations form a positive proportion of allpermutations (asymptotically 1/e2)
◮ In many (conjecturally all) proper permutation classesthey have density 0
◮ We can hope to understand a class by understandingits simples and how they inflate
◮ Specifically, this may yield functional equations of thegenerating function and hence computations of theenumeration and/or growth rate
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Finitely many simple permutations
TheoremIf a class has only finitely many simple permutations then it hasan algebraic generating function.
◮ A and Atkinson (2005)◮ Effective ‘in principle’, i.e. an algorithm for computing a
defining system of equations for the generating function◮ Some interesting corollaries, e.g. if a class has finitely
many simples and does not contain arbitrarily longdecreasing permutations then it has a rational generatingfunction
◮ “The prime reason for giving this example is to show thatwe are not necessarily stymied if the number of simplepermutations is infinite.”
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Encodings
◮ The enumerative combinatorics of words and trees isbetter understood
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Encodings
◮ The enumerative combinatorics of words and trees isbetter understood
◮ So, apply leverage from there by encoding permutationclasses
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Encodings
◮ The enumerative combinatorics of words and trees isbetter understood
◮ So, apply leverage from there by encoding permutationclasses
◮ For trees this approach goes by many names
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Encodings
◮ The enumerative combinatorics of words and trees isbetter understood
◮ So, apply leverage from there by encoding permutationclasses
◮ For trees this approach goes by many names◮ For words, A, At and Ruškuc (2003) give (fairly) general
criteria for the encoding of a permutation class by a regularlanguage
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Encodings
◮ The enumerative combinatorics of words and trees isbetter understood
◮ So, apply leverage from there by encoding permutationclasses
◮ For trees this approach goes by many names◮ For words, A, At and Ruškuc (2003) give (fairly) general
criteria for the encoding of a permutation class by a regularlanguage
◮ Extended to a new type of encoding, the insertionencoding (prefigured by Viennot) by A, Linton and R (2005)
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Grid classes
◮ The notion of griddable class was central to V’scharacterization of small permutation classes
◮ Loosely, a griddable class is associated with a matrixwhose entries are (simpler) permutation classes
◮ All permutations in the class can be chopped apart intosections that correspond to the matrix entries
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Geometric monotone grid classes
In a geometric grid class, the permutations need to be drawnfrom the points of a particular representation in R
2
Theorem (A, At, Bouvel, R and V (to appear TAMS))Every geometrically griddable class:
◮ is partially well ordered;◮ is finitely based;◮ is in bijection with a regular language and thus has a
rational generating function.
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
◮ Let C[D] represent the inflations of permutations in C bypermutations in D, and let 〈C〉 denote the closure of Cunder inflation
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
◮ Let C[D] represent the inflations of permutations in C bypermutations in D, and let 〈C〉 denote the closure of Cunder inflation
◮ If C is geometrically griddable, then every subclass of 〈C〉 isfinitely based and partially well ordered
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
◮ Let C[D] represent the inflations of permutations in C bypermutations in D, and let 〈C〉 denote the closure of Cunder inflation
◮ If C is geometrically griddable, then every subclass of 〈C〉 isfinitely based and partially well ordered
◮ If C is geometrically griddable, then every subclass of 〈C〉has an algebraic generating function
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
◮ Let C[D] represent the inflations of permutations in C bypermutations in D, and let 〈C〉 denote the closure of Cunder inflation
◮ If C is geometrically griddable, then every subclass of 〈C〉 isfinitely based and partially well ordered
◮ If C is geometrically griddable, then every subclass of 〈C〉has an algebraic generating function
◮ If C is geometrically griddable and U is strongly rational,then C[U ] is also strongly rational
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Beyond grid classes
Results from Inflations of Geometric Grid Classes ofPermutations, A, R and V (arxiv.org/abs/1202.1833):
◮ Let C[D] represent the inflations of permutations in C bypermutations in D, and let 〈C〉 denote the closure of Cunder inflation
◮ If C is geometrically griddable, then every subclass of 〈C〉 isfinitely based and partially well ordered
◮ If C is geometrically griddable, then every subclass of 〈C〉has an algebraic generating function
◮ If C is geometrically griddable and U is strongly rational,then C[U ] is also strongly rational
◮ Every small permutation class has a rational generatingfunction
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Quasi-applications
◮ These ideas, together with a certain amount of numbereight wire or duct tape, i.e. “un peu de rafistolage” can beused to compute enumerations for some (arguably)interesting classes
◮ Examples from the basic environment of geometric gridclasses are considered in A, At and Brignall: TheEnumeration of Three Pattern Classes using MonotoneGrid Classes (EJC 19.3 (2012) P20)
◮ Examples for inflations of geometric grid classes areconsidered in A, At and V: Inflations of Geometric GridClasses: Three Case Studies (arxiv.org/abs/1209.0425)
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Av(4312, 3142)
◮ Every simple permutation in this class lies in the geometricgrid class:
◮ This yields a regular language for the simple permutations◮ The allowed inflations of these permutations are easily
described, yielding a recursive description of the class◮ This leads to an equation for its generating function:
(x3 − 2x2 + x)f 4 + (4x3 − 9x2 + 6x − 1)f 3
+ (6x3 − 12x2 + 7x − 1)f 2
+ (4x3 − 5x2 + x)f+ x3 = 0
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Av(321)This class is in some sense a limit of geometric grid classes:
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Av(321)This class is in some sense a limit of geometric grid classes:
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Av(321)This class is in some sense a limit of geometric grid classes:
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Av(321)This class is in some sense a limit of geometric grid classes:
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Av(321)This class is in some sense a limit of geometric grid classes:
![Page 48: Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It seems plausible that the fewer permutations there are in a class, or alternatively](https://reader034.fdocument.pub/reader034/viewer/2022050220/5f66348028062b77f37baf59/html5/thumbnails/48.jpg)
Av(321)This class is in some sense a limit of geometric grid classes:
![Page 49: Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It seems plausible that the fewer permutations there are in a class, or alternatively](https://reader034.fdocument.pub/reader034/viewer/2022050220/5f66348028062b77f37baf59/html5/thumbnails/49.jpg)
Av(321)This class is in some sense a limit of geometric grid classes:
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Av(321)
ConjectureEvery finitely based proper subclass of Av(321) has a rationalgenerating function
This class is in some sense a limit of geometric grid classes:
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Av(4231, 35142, 42513, 351624)
◮ Enumerating this class was mentioned as a challengeproblem by Alexander Woo at Permutation Patterns 2012
◮ These permutations index “Schubert varieties defined byinclusions”
◮ The simple permutations look like
◮ Somehow, this leads to an enumeration of the class (acomplicated rational function in
√1 − 4x)
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Where to from here?
◮ Underlying the main results on geometric grid classes andtheir inflations is a notion of natural encoding, in this caseof permutations by words, which may be applicable toother types of combinatorial structure particularly thosecarrying a linear order
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Where to from here?
◮ Underlying the main results on geometric grid classes andtheir inflations is a notion of natural encoding, in this caseof permutations by words, which may be applicable toother types of combinatorial structure particularly thosecarrying a linear order
◮ Understanding the limit case for geometric grid classesbetter
![Page 54: Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It seems plausible that the fewer permutations there are in a class, or alternatively](https://reader034.fdocument.pub/reader034/viewer/2022050220/5f66348028062b77f37baf59/html5/thumbnails/54.jpg)
Where to from here?
◮ Underlying the main results on geometric grid classes andtheir inflations is a notion of natural encoding, in this caseof permutations by words, which may be applicable toother types of combinatorial structure particularly thosecarrying a linear order
◮ Understanding the limit case for geometric grid classesbetter
◮ Understanding non-geometric grid classes (David Bevan, astudent of R. Brignall has some nice results in the casewhere there is only one cycle, and is working on moregeneral cases)
![Page 55: Nouvelles perspectives sur l'énumération des classes de ... · 07/09/2012 · First steps It seems plausible that the fewer permutations there are in a class, or alternatively](https://reader034.fdocument.pub/reader034/viewer/2022050220/5f66348028062b77f37baf59/html5/thumbnails/55.jpg)
Where to from here?
◮ Underlying the main results on geometric grid classes andtheir inflations is a notion of natural encoding, in this caseof permutations by words, which may be applicable toother types of combinatorial structure particularly thosecarrying a linear order
◮ Understanding the limit case for geometric grid classesbetter
◮ Understanding non-geometric grid classes (David Bevan, astudent of R. Brignall has some nice results in the casewhere there is only one cycle, and is working on moregeneral cases)
◮ Permutation Patterns 2013: July 1-5, Paris