Pattern-Avoiding Permutations and Lattice Paths: Old Connections and - PowerPoint PPT Presentation

Pattern-Avoiding Permutations and Lattice Paths: Old Connections and New Links Eric S. Egge Carleton College August 3, 2012 Eric S. Egge (Carleton College) Pattern-Avoiding Permutations and Lattice Paths: Old Connections and New Links August 3, 2012 1 / 12

Permutations and Pattern Avoidance Definition π, σ are permutations. π avoids σ whenever π has no subsequence with same length and relative order as σ . Example 6152347 avoids 231 but not 213. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 2 / 12

Permutations and Pattern Avoidance Definition π, σ are permutations. π avoids σ whenever π has no subsequence with same length and relative order as σ . Example 6152347 avoids 231 but not 213. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 2 / 12

Permutations and Pattern Avoidance Definition t t π, σ are permutations. t π avoids σ whenever π has no subsequence t with same length and relative order as σ . t t t Example The diagram of 6152347. 6152347 avoids 231 but not 213. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 2 / 12

Permutations and Pattern Avoidance Definition t t π, σ are permutations. t π avoids σ whenever π has no subsequence t with same length and relative order as σ . t t t Example The diagram of 6152347. 6152347 avoids 231 but not 213. t t t Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 2 / 12

Permutations and Pattern Avoidance Definition t t π, σ are permutations. t π avoids σ whenever π has no subsequence t with same length and relative order as σ . t t t Example The diagram of 6152347. 6152347 avoids 231 but not 213. t t Notation t Av ( σ ) := set of all permutations which avoid σ . Av n ( σ ) = Av ( σ ) ∩ S n Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 2 / 12

Counting Pattern-Avoiding Permutations | Av n (132) | = | Av n (213) | = | Av n (231) | = | Av n (312) | t t t t t t t t t t t t | Av n (321) | = | Av n (123) | t t t t t t Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 3 / 12

Counting Pattern-Avoiding Permutations | Av n (132) | = | Av n (213) | = | Av n (231) | = | Av n (312) | t t t t t t t t t t t t | Av n (321) | = | Av n (123) | Idea Rotation of diagrams gives bijections t t among these sets. t t t t Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 3 / 12

Counting Pattern-Avoiding Permutations | Av n (132) | = | Av n (213) | = | Av n (231) | = | Av n (312) | t t t t t t t t t t t t | Av n (321) | = | Av n (123) | Idea Rotation of diagrams gives bijections t t among these sets. t t t t Theorem 1 � 2 n � | Av n (231) | = | Av n (321) | = C n = n + 1 n Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 3 / 12

Catalan Paths Definition A Catalan path (of length n ) is a sequence of n North (0 , 1) steps and n East (1 , 0) steps which never passes below the line y = x . Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 4 / 12

Catalan Paths � � Definition � � A Catalan path (of length n ) is a � sequence of n North (0 , 1) steps and n � � East (1 , 0) steps which never passes � below the line y = x . � � Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 4 / 12

Catalan Paths � � Definition � � A Catalan path (of length n ) is a � sequence of n North (0 , 1) steps and n � � East (1 , 0) steps which never passes � below the line y = x . � � Theorem The number of Catalan paths of length n is � 2 n � 1 C n = . n + 1 n Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 4 / 12

Recursive Structures Permutations t t t t t t t t 12438756 avoids 231. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations t t t t t t t t 12438756 avoids 231. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations t π 2 π 1 π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations Paths � � t � � π 2 � � � � π 1 � � π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations Paths � � t � � π 2 � � � � π 1 � � π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations Paths � t � � π 1 π 2 � � � � π 2 � π 1 � π 1 ⊕ π 2 π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Recursive Structures Permutations Paths � t � � π 1 π 2 � � � � π 2 � π 1 � π 1 ⊕ π 2 π 1 ⊕ π 2 Idea F ( π 1 ⊕ π 2 ) = N F ( π 2 ) E F ( π 1 ) Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 5 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t π 2 π 1 π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t t t π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t t π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t t t π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t � � � π 2 � � � � π 1 � � � π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions Definition An inversion in a permutation is an occurence of the pattern 21. Theorem inv( π 1 ⊕ π 2 ) = inv( π 1 ) + inv( π 2 ) + length( π 2 ) t � � � � � � π 2 � � � � � � � � π 1 � � � � � π 1 ⊕ π 2 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 6 / 12

Bonus Information: Inversions and Area Definition The area of a lattice path π is the number of full squares below π and above y = x . Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 7 / 12

Bonus Information: Inversions and Area Definition The area of a lattice path π is the number of full squares below π and above y = x . Theorem area( π 1 ⊕ π 2 ) = area( π 1 ) + area( π 2 ) + length( π 2 ) Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 7 / 12

Bonus Information: Inversions and Area Definition The area of a lattice path π is the number of full squares below π and above y = x . Theorem area( π 1 ⊕ π 2 ) = area( π 1 ) + area( π 2 ) + length( π 2 ) Theorem inv( π ) = area( F ( π )) Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 7 / 12

A Bonus Bonus Definition For any permutation π and number k , let k ( π ) be the number of decreasing subsequences of length k in π . Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 8 / 12

A Bonus Bonus Definition For any permutation π and number k , let k ( π ) be the number of decreasing subsequences of length k in π . Definition The height ht( s ) of an East step s in a Catalan path π is the number of � ht( s ) � area squares below it. The kth area of π is area k ( π ) = � . s ∈ π k − 1 Eric S. Egge (Carleton College) Permutations and Lattice Paths August 3, 2012 8 / 12