Simple Permutations R.L.F. Brignall joint work with Sophie - PowerPoint PPT Presentation

Simple Permutations R.L.F. Brignall joint work with Sophie Huczynska, Nik Ruškuc and Vincent Vatter School of Mathematics and Statistics University of St Andrews Thursday 15th June, 2006

Introduction Basic Concepts 1 Permutation Classes Intervals and Simple Permutations Algebraic Generating Functions for Sets of Permutations 2 Finitely Many Simples Sets of Permutations A Decomposition Theorem with Enumerative Consequences 3 Aim Pin Sequences Decomposing Simple Permutations Decidability and Unavoidable Structures 4 More on Pins Decidability

Basic Concepts Outline Basic Concepts 1 Permutation Classes Intervals and Simple Permutations Algebraic Generating Functions for Sets of Permutations 2 Finitely Many Simples Sets of Permutations A Decomposition Theorem with Enumerative Consequences 3 Aim Pin Sequences Decomposing Simple Permutations Decidability and Unavoidable Structures 4 More on Pins Decidability

Basic Concepts Permutation Classes Pattern Involvement Regard a permutation of length n as an ordering of the symbols 1 , . . . , n . A permutation τ = t 1 t 2 . . . t k is involved in the permutation σ = s 1 s 2 . . . s n if there exists a subsequence s i 1 , s i 2 , . . . , s i k order isomorphic to τ . Example

Basic Concepts Permutation Classes Pattern Involvement Regard a permutation of length n as an ordering of the symbols 1 , . . . , n . A permutation τ = t 1 t 2 . . . t k is involved in the permutation σ = s 1 s 2 . . . s n if there exists a subsequence s i 1 , s i 2 , . . . , s i k order isomorphic to τ . Example

Basic Concepts Permutation Classes Pattern Involvement Regard a permutation of length n as an ordering of the symbols 1 , . . . , n . A permutation τ = t 1 t 2 . . . t k is involved in the permutation σ = s 1 s 2 . . . s n if there exists a subsequence s i 1 , s i 2 , . . . , s i k order isomorphic to τ . Example 1 3 5 2 4 4 2 1 6 3 8 5 7

Basic Concepts Permutation Classes Pattern Involvement Regard a permutation of length n as an ordering of the symbols 1 , . . . , n . A permutation τ = t 1 t 2 . . . t k is involved in the permutation σ = s 1 s 2 . . . s n if there exists a subsequence s i 1 , s i 2 , . . . , s i k order isomorphic to τ . Example 1 3 5 2 4 4 2 1 6 3 8 5 7 ≺

Basic Concepts Permutation Classes Permutation Classes Involvement forms a partial order on the set of all permutations. Downsets of permutations in this partial order form permutation classes. A permutation class C can be seen to avoid certain permutations. Write C = Av ( B ) . Example The class C = Av ( 12 ) consists of all the decreasing permutations: { 1 , 21 , 321 , 4321 , . . . }

Basic Concepts Permutation Classes Permutation Classes Involvement forms a partial order on the set of all permutations. Downsets of permutations in this partial order form permutation classes. A permutation class C can be seen to avoid certain permutations. Write C = Av ( B ) . Example The class C = Av ( 12 ) consists of all the decreasing permutations: { 1 , 21 , 321 , 4321 , . . . }

Basic Concepts Permutation Classes Permutation Classes Involvement forms a partial order on the set of all permutations. Downsets of permutations in this partial order form permutation classes. A permutation class C can be seen to avoid certain permutations. Write C = Av ( B ) . Example The class C = Av ( 12 ) consists of all the decreasing permutations: { 1 , 21 , 321 , 4321 , . . . }

Basic Concepts Permutation Classes Permutation Classes Involvement forms a partial order on the set of all permutations. Downsets of permutations in this partial order form permutation classes. A permutation class C can be seen to avoid certain permutations. Write C = Av ( B ) . Example The class C = Av ( 12 ) consists of all the decreasing permutations: { 1 , 21 , 321 , 4321 , . . . }

Basic Concepts Permutation Classes Permutation Classes Involvement forms a partial order on the set of all permutations. Downsets of permutations in this partial order form permutation classes. A permutation class C can be seen to avoid certain permutations. Write C = Av ( B ) . Example The class C = Av ( 12 ) consists of all the decreasing permutations: { 1 , 21 , 321 , 4321 , . . . }

Basic Concepts Permutation Classes Generating Functions C n – permutations in C of length n . |C n | x n is the generating function. � Example The generating function of C = Av ( 12 ) is: 1 1 + x + x 2 + x 3 + · · · = 1 − x

Basic Concepts Permutation Classes Generating Functions C n – permutations in C of length n . |C n | x n is the generating function. � Example The generating function of C = Av ( 12 ) is: 1 1 + x + x 2 + x 3 + · · · = 1 − x

Basic Concepts Permutation Classes Generating Functions C n – permutations in C of length n . |C n | x n is the generating function. � Example The generating function of C = Av ( 12 ) is: 1 1 + x + x 2 + x 3 + · · · = 1 − x

Basic Concepts Permutation Classes Generating Functions C n – permutations in C of length n . |C n | x n is the generating function. � Example The generating function of C = Av ( 12 ) is: 1 1 + x + x 2 + x 3 + · · · = 1 − x

Basic Concepts Intervals and Simple Permutations Intervals Pick any permutation π . An interval of π is a set of contiguous indices I = [ a , b ] such that π ( I ) = { π ( i ) : i ∈ I } is also contiguous. Example

Basic Concepts Intervals and Simple Permutations Intervals Pick any permutation π . An interval of π is a set of contiguous indices I = [ a , b ] such that π ( I ) = { π ( i ) : i ∈ I } is also contiguous. Example

Basic Concepts Intervals and Simple Permutations Intervals Pick any permutation π . An interval of π is a set of contiguous indices I = [ a , b ] such that π ( I ) = { π ( i ) : i ∈ I } is also contiguous. Example

Basic Concepts Intervals and Simple Permutations Intervals Pick any permutation π . An interval of π is a set of contiguous indices I = [ a , b ] such that π ( I ) = { π ( i ) : i ∈ I } is also contiguous. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Simple Permutations Only intervals are singletons and the whole thing. Example

Basic Concepts Intervals and Simple Permutations Special Simple Permutations Parallel alternations. Wedge permutations Two flavours of wedge simple permutation.

Basic Concepts Intervals and Simple Permutations Special Simple Permutations Parallel alternations. Wedge permutations Two flavours of wedge simple permutation.

Basic Concepts Intervals and Simple Permutations Special Simple Permutations Parallel alternations. Wedge permutations – not simple! Two flavours of wedge simple permutation.

Basic Concepts Intervals and Simple Permutations Special Simple Permutations Parallel alternations. Wedge permutations Two flavours of wedge simple permutation.

Algebraic Generating Functions for Sets of Permutations Outline Basic Concepts 1 Permutation Classes Intervals and Simple Permutations Algebraic Generating Functions for Sets of Permutations 2 Finitely Many Simples Sets of Permutations A Decomposition Theorem with Enumerative Consequences 3 Aim Pin Sequences Decomposing Simple Permutations Decidability and Unavoidable Structures 4 More on Pins Decidability

Algebraic Generating Functions for Sets of Permutations Finitely Many Simples Motivation Example Av ( 132 ) Av ( 132 ) 132-avoiders: generic structure. Only simple permutations are1, 12, and 21. Enumerate recursively: f ( x ) = xf ( x ) 2 + 1.

Algebraic Generating Functions for Sets of Permutations Finitely Many Simples Motivation Example Av ( 132 ) Av ( 132 ) 132-avoiders: generic structure. Only simple permutations are1, 12, and 21. Enumerate recursively: f ( x ) = xf ( x ) 2 + 1.

Algebraic Generating Functions for Sets of Permutations Finitely Many Simples Motivation Example Av ( 132 ) Av ( 132 ) 132-avoiders: generic structure. Only simple permutations are1, 12, and 21. Enumerate recursively: f ( x ) = xf ( x ) 2 + 1.