Enumeration of Pin-Permutations Fr´ ed´ erique Bassino Mathilde Bouvel Dominique Rossin ees AL´ Journ´ EA 2009 liafa
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Main result of the talk Conjecture [Brignall, Ruˇ skuc, Vatter] : The pin-permutation class has a rational generating function. Theorem: The generating function of the pin-permutation class is 8 z 6 − 20 z 5 − 4 z 4 + 12 z 3 − 9 z 2 + 6 z − 1 P ( z ) = z 8 z 8 − 20 z 7 + 8 z 6 + 12 z 5 − 14 z 4 + 26 z 3 − 19 z 2 + 8 z − 1 Technique for the proof: Characterize the decomposition trees of pin-permutations Compute the generating function of simple pin-permutations Put things together to compute the generating function of pin-permutations Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Outline of the talk 1 Introduction: permutation classes 2 Definition of pin-permutations 3 Substitution decomposition and decomposition trees 4 Characterization of the decomposition trees of pin-permutations 5 Generating function of the pin-permutation class 6 Conclusion and discussion on the basis Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Representations of permutations Permutation: Bijective map from [1 .. n ] to itself Graphical representation: One-line representation: σ = 1 8 3 6 4 2 5 7 Two-line representation: � 1 2 3 4 5 6 7 8 � σ = 1 8 3 6 4 2 5 7 Cyclic representation: σ = (1) (2 8 7 5 4 6) (3) Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Patterns in permutations Pattern relation � : π ∈ S k is a pattern of σ ∈ S n when ∃ 1 ≤ i 1 < . . . < i k ≤ n such that σ i 1 . . . σ i k is order-isomorphic to π . We write π � σ . Equivalently : Normalizing σ i 1 . . . σ i k on [1 .. k ] yields π . Example: 1 2 3 4 � 3 1 2 8 5 4 7 9 6 since 1 2 5 7 ≡ 1 2 3 4. Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Patterns in permutations Pattern relation � : π ∈ S k is a pattern of σ ∈ S n when ∃ 1 ≤ i 1 < . . . < i k ≤ n such that σ i 1 . . . σ i k is order-isomorphic to π . We write π � σ . Equivalently : Normalizing σ i 1 . . . σ i k on [1 .. k ] yields π . Example: 1 2 3 4 � 3 1 2 8 5 4 7 9 6 since 1 2 5 7 ≡ 1 2 3 4. Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Classes of permutations Class of permutations: set downward closed for � Equivalently : σ ∈ C and π � σ ⇒ π ∈ C S ( B ): the class of perm. avoiding all the patterns in the basis B . Prop.: Every class C is characterized by its basis: C = S ( B ) for B = { σ / ∈ C : ∀ π � σ with π � = σ, π ∈ C} Basis may be finite or infinite. Enumeration [Stanley-Wilf, Marcus-Tardos] : | S n ( B ) | ≤ c n B Two points of view: class given by its basis or by a (graphical) property stable for � Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Classes of permutations Class of permutations: set downward closed for � Equivalently : σ ∈ C and π � σ ⇒ π ∈ C S ( B ): the class of perm. avoiding all the patterns in the basis B . Prop.: Every class C is characterized by its basis: C = S ( B ) for B = { σ / ∈ C : ∀ π � σ with π � = σ, π ∈ C} Basis may be finite or infinite. Enumeration [Stanley-Wilf, Marcus-Tardos] : | S n ( B ) | ≤ c n B Two points of view: class given by its basis or by a (graphical) property stable for � Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Introduction: permutation classes Simple permutations Interval = window of elements of σ whose values form a range Example: 5 7 4 6 is an interval of 2 5 7 4 6 1 3 Simple permutation = has no interval except 1 , 2 , . . . , n and σ Example: 3 1 7 4 6 2 5 is simple. Smallest ones : 1 2 , 2 1 , 2 4 1 3 , 3 1 4 2 Decomposition trees: formalize the idea that simple permutations are “building blocks” for all permutations Thm [Albert Atkinson] : C contains finitely many simple permutations ⇒ C has an algebraic generating function Pin-permutations: used for deciding whether C contains finitely many simple permutations Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p i − 1 p i • the separation condition p 1 . . . p i − 2 p i • or the independence condition = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p i − 1 p i • the separation condition p 1 . . . p i − 2 p i • or the independence condition p 1 = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p i − 1 p i • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p 3 p i − 1 p i • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p 3 p i − 1 p i p 4 • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p 3 p i − 1 p i p 4 • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 p 5 = bounding box of { p 1 , . . . , p i − 1 } Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p i the externality condition and p 3 p i − 1 p i p 4 • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 p 5 = bounding box of { p 1 , . . . , p i − 1 } p 6 Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p 7 p i the externality condition and p 3 p i − 1 p i p 4 • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 p 5 = bounding box of { p 1 , . . . , p i − 1 } p 6 Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Pin representations Pin representation of σ = sequence Example: ( p 1 , . . . , p n ) such that each p i satisfies p 7 p i p 8 the externality condition and p 3 p i − 1 p i p 4 • the separation condition p 1 . . . p i − 2 p i p 2 • or the independence condition p 1 p 5 = bounding box of { p 1 , . . . , p i − 1 } p 6 Mathilde Bouvel Pin-Permutations
Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Definition of pin-permutations Non-uniqueness of pin representation p 7 p 8 p 8 p 6 p 3 p 1 p 4 p 3 p 2 p 2 p 1 p 5 p 5 p 4 p 6 p 7 Mathilde Bouvel Pin-Permutations
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