The neighbours of Baxter numbers Lattice paths Veronica Guerrini - PowerPoint PPT Presentation

Number sequences Generating trees Slicings of parallelogram polyominoes Slicings generalizations Permutations Semi-Baxter sequence The neighbours of Baxter numbers Lattice paths Veronica Guerrini University of Siena, DIISM 31 January 2017, LIPN, Paris

Catalan sequence: 1,2,5,14,42,...(A000108) Number sequences Dyck paths, AV (132),... Generating trees Slicings of parallelogram polyominoes Slicings generalizations Permutations Baxter sequence: Semi-Baxter 1,2,6,22,92,...(A001181) sequence Lattice paths AV (2-41-3 , 3-14-2),... Factorial sequence: 1,2,6,24,120,...(A000142) permutations,...

Catalan sequence: Goal 1. 1,2,5,14,42,...(A000108) Number To provide a sequences Dyck paths, AV (132),... continuum from Generating trees Catalan to Schröder sequence: Slicings of parallelogram Baxter through polyominoes 1,2,6,22,90,...(A006318) Schröder. Slicings Schröder paths, separable generalizations permutations,... Permutations Baxter sequence: Semi-Baxter 1,2,6,22,92,...(A001181) sequence Lattice paths AV (2-41-3 , 3-14-2),... Factorial sequence: 1,2,6,24,120,...(A000142) permutations,...

Catalan sequence: Goal 1. 1,2,5,14,42,...(A000108) Number To provide a sequences Dyck paths, AV (132),... continuum from Generating trees Catalan to Schröder sequence: Slicings of parallelogram Baxter through polyominoes 1,2,6,22,90,...(A006318) Schröder. Slicings Schröder paths, separable generalizations permutations,... Permutations Baxter sequence: Semi-Baxter 1,2,6,22,92,...(A001181) sequence Lattice paths AV (2-41-3 , 3-14-2),... Goal 2. To provide a Semi-Baxter sequence: continuum from 1,2,6,23,104,...(A117106) Baxter to Factorial plane permutations, through semi-Baxter. AV (2-41-3),... Factorial sequence: 1,2,6,24,120,...(A000142) permutations,...

How to establish such continuum? Number sequences Generating trees At the abstract level of generating trees and succession rules so that Slicings of each inclusion is valid for all the families of objects enumerated by parallelogram polyominoes the corresponding sequences. Slicings generalizations ECO method. Enumerating Combinatorial Objects is a method for Permutations the exhaustive generation of a class C of combinatorial objects Semi-Baxter sequence equipped with a size | · | : C → N . Lattice paths An ECO-operator is ϑ : C n → 2 C n +1 s.t. - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 .

How to establish such continuum? Number sequences At the abstract level of generating trees and succession rules so that Generating trees each inclusion is valid for all the families of objects enumerated by Slicings of the corresponding sequences. parallelogram polyominoes ECO method. Enumerating Combinatorial Objects is a method for Slicings generalizations the exhaustive generation of a class C of combinatorial objects Permutations equipped with a size | · | : C → N . Semi-Baxter sequence An ECO-operator is ϑ : C n → 2 C n +1 s.t. Lattice paths - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 . A permutation π of length n avoids τ of length k ≤ n iff there are no i 1 , . . . , i k such that π i 1 . . . π i k is order isomorphic to τ . Example. π = 6 4 2 1 5 3 contains τ = 1 3 2; ρ = 6 4 3 5 1 2 avoids τ .

How to establish such continuum? Number sequences At the abstract level of generating trees and succession rules so that Generating trees each inclusion is valid for all the families of objects enumerated by Slicings of the corresponding sequences. parallelogram polyominoes ECO method. Enumerating Combinatorial Objects is a method for Slicings generalizations the exhaustive generation of a class C of combinatorial objects Permutations equipped with a size | · | : C → N . Semi-Baxter sequence An ECO-operator is ϑ : C n → 2 C n +1 s.t. Lattice paths - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 . A permutation π of length n avoids τ of length k ≤ n iff there are no i 1 , . . . , i k such that π i 1 . . . π i k is order isomorphic to τ . Example. π = 6 4 2 1 5 3 contains τ = 1 3 2; ρ = 6 4 3 5 1 2 avoids τ .

How to establish such continuum? Number sequences At the abstract level of generating trees and succession rules so that Generating trees each inclusion is valid for all the families of objects enumerated by Slicings of the corresponding sequences. parallelogram polyominoes ECO method. Enumerating Combinatorial Objects is a method for Slicings generalizations the exhaustive generation of a class C of combinatorial objects Permutations equipped with a size | · | : C → N . Semi-Baxter sequence An ECO-operator is ϑ : C n → 2 C n +1 s.t. Lattice paths - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 . A permutation π of length n avoids τ of length k ≤ n iff there are no i 1 , . . . , i k such that π i 1 . . . π i k is order isomorphic to τ . Example. π = 6 4 2 1 5 3 contains τ = 1 3 2; ρ = 6 4 3 5 1 2 avoids τ .

How to establish such continuum? Number sequences At the abstract level of generating trees and succession rules so that Generating trees each inclusion is valid for all the families of objects enumerated by Slicings of the corresponding sequences. parallelogram polyominoes ECO method. Enumerating Combinatorial Objects is a method for Slicings generalizations the exhaustive generation of a class C of combinatorial objects Permutations equipped with a size | · | : C → N . Semi-Baxter sequence An ECO-operator is ϑ : C n → 2 C n +1 s.t. Lattice paths - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 . A permutation π of length n avoids τ of length k ≤ n iff there are no i 1 , . . . , i k such that π i 1 . . . π i k is order isomorphic to τ . Example. π = 6 4 2 1 5 3 contains τ = 1 3 2; ρ = 6 4 3 5 1 2 avoids τ .

How to establish such continuum? Number sequences At the abstract level of generating trees and succession rules so that Generating trees each inclusion is valid for all the families of objects enumerated by Slicings of the corresponding sequences. parallelogram polyominoes ECO method. Enumerating Combinatorial Objects is a method for Slicings generalizations the exhaustive generation of a class C of combinatorial objects Permutations equipped with a size | · | : C → N . Semi-Baxter sequence An ECO-operator is ϑ : C n → 2 C n +1 s.t. Lattice paths - for any o , o ′ ∈ C n , if o � = o ′ , then ϑ ( o ) ∩ ϑ ( o ′ ) = ∅ ; - � o ∈C n ϑ ( o ) = C n +1 . A permutation π of length n avoids τ of length k ≤ n iff there are no i 1 , . . . , i k such that π i 1 . . . π i k is order isomorphic to τ . Example. π = 6 4 2 1 5 3 contains τ = 1 3 2; ρ = 6 4 3 5 1 2 avoids τ .

How to establish such continuum? Number sequences Definition. Generating trees Slicings of Let ϑ be an ECO-operator for C . A generating tree for C is a infinite parallelogram polyominoes rooted tree such that the vertices at level n are the objects of size n Slicings and their sons are the objects produced by ϑ . generalizations Permutations Semi-Baxter 1 sequence 1 2 2 1 Lattice paths 1 2 3 2 3 1 2 1 3 3 1 2 3 2 1 A compact notation for generating trees is the notion of: Definition. A succession rule is system (( r ) , S ) consisting of an axiom ( r ) and a set of productions S � ( r ) Ω = ( ℓ ) � ( e 1 ) , ( e 2 ) , . . . , ( e k ( ℓ ) )

How to establish such continuum? Number sequences Definition. Generating trees Slicings of Let ϑ be an ECO-operator for C . A generating tree for C is a infinite parallelogram polyominoes rooted tree such that the vertices at level n are the objects of size n Slicings and their sons are the objects produced by ϑ . generalizations Permutations (1) Semi-Baxter sequence (1) (2) Lattice paths (1) (2) (1) (2) (3) A compact notation for generating trees is the notion of: Definition. A succession rule is system (( r ) , S ) consisting of an axiom ( r ) and a set of productions S � ( r ) Ω = ( ℓ ) � ( e 1 ) , ( e 2 ) , . . . , ( e k ( ℓ ) )

How to establish such continuum? Number sequences Definition. Generating trees Slicings of Let ϑ be an ECO-operator for C . A generating tree for C is a infinite parallelogram polyominoes rooted tree such that the vertices at level n are the objects of size n Slicings and their sons are the objects produced by ϑ . generalizations Permutations Semi-Baxter sequence � (1) Lattice paths Ω Cat = ( i ) � (1) , (2) , . . . , ( i ) , ( i + 1) A compact notation for generating trees is the notion of: Definition. A succession rule is system (( r ) , S ) consisting of an axiom ( r ) and a set of productions S � ( r ) Ω = ( ℓ ) � ( e 1 ) , ( e 2 ) , . . . , ( e k ( ℓ ) )

Examples Number sequences Generating trees Slicings of Catalan succession rule: parallelogram polyominoes � (1) Slicings Ω Cat = generalizations ( i ) � (1) , (2) , . . . , ( i ) , ( i + 1) Permutations Semi-Baxter sequence Schröder succession rule: Lattice paths � (2) Ω Sep = ( j ) � (2) , (3) , . . . , ( j ) , ( j + 1) , ( j + 1) Baxter succession rule:  (1 , 1)  Ω Bax = ( h , k ) � (1 , k + 1) , . . . , ( h , k + 1) ( h + 1 , 1) , . . . , ( h + 1 , k ) 