Classical pattern distribution in S n (132) and S n (123) Dun Qiu - PowerPoint PPT Presentation

Classical pattern distribution in S n (132) and S n (123) Dun Qiu UC San Diego duqiu@ucsd.edu Based on joint work with Jeffrey Remmel Permutation Patterns 2018, Dartmouth College July 11, 2018 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 1 / 28

In Memory of Jeffrey Remmel Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 2 / 28

Outline Motivation 1 Introduction 2 Wilf-equivalence of Q γ λ ( t , x ) 3 Recursions of Q γ λ ( t , x ) 4 Other Results and Open Problems 5 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 3 / 28

Outline Motivation 1 Introduction 2 Wilf-equivalence of Q γ λ ( t , x ) 3 Recursions of Q γ λ ( t , x ) 4 Other Results and Open Problems 5 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 3 / 28

Motivation Ran Pan’s Project P http://www.math.ucsd.edu/ ∼ projectp/ Problem 13: enumerate permutations in S n avoiding a classical pattern and a consecutive pattern at the same time. Then Professor Remmel conducted researchs on distribution of classical patterns and consecutive patterns in S n (132) and S n (123). Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 4 / 28

Outline Motivation 1 Introduction 2 Wilf-equivalence of Q γ λ ( t , x ) 3 Recursions of Q γ λ ( t , x ) 4 Other Results and Open Problems 5 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 4 / 28

Permutations, Descents, LRmins A permutation σ = σ 1 · · · σ n of [ n ] = { 1 , . . . , n } is a rearrangement of the numbers 1 , . . . , n . The set of permutations of [ n ] is denoted by S n . σ i is a descent if σ i > σ i +1 . des ( σ ) is the number of descents in σ . We let LRmin ( σ ) denote the number of left to right minima of σ . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 5 / 28

Inversions, Coinversions ( σ i , σ j ) is an inversion if i < j and σ i > σ j . inv ( σ ) denotes the number of inversions in σ . ( σ i , σ j ) is a coinversion if i < j and σ i < σ j . coinv ( σ ) denotes the number of coinversions in σ . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 6 / 28

Reduction of A Sequence Given a sequence of distinct positive integers w = w 1 . . . w n , we let the reduction (or standardization) of the sequence, red ( w ), denote the permutation of [ n ] obtained from w by replacing the i -th smallest letter in w by i . Example If w = 4592, then red ( w ) = 2341. Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 7 / 28

Classical Patterns Occurrence and Avoidance Given a permutation τ = τ 1 . . . τ j in S j , we say the pattern τ occurs in σ = σ 1 . . . σ n ∈ S n if there exist 1 ≤ i 1 < · · · < i j ≤ n such that red ( σ i 1 . . . σ i j ) = τ . We let occr τ ( σ ) denote the number of τ occurrence in σ . We say σ avoids the pattern τ if τ does not occur in σ . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 8 / 28

Classical Patterns Occurrence and Avoidance Given a permutation τ = τ 1 . . . τ j in S j , we say the pattern τ occurs in σ = σ 1 . . . σ n ∈ S n if there exist 1 ≤ i 1 < · · · < i j ≤ n such that red ( σ i 1 . . . σ i j ) = τ . We let occr τ ( σ ) denote the number of τ occurrence in σ . We say σ avoids the pattern τ if τ does not occur in σ . Example π = 867932451 avoids pattern 132, contains pattern 123. occr 123 ( π ) = 2 since pattern occurrences are 6 , 7 , 9 and 3 , 4 , 5. Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 8 / 28

Classical Patterns Occurrence and Avoidance Given a permutation τ = τ 1 . . . τ j in S j , we say the pattern τ occurs in σ = σ 1 . . . σ n ∈ S n if there exist 1 ≤ i 1 < · · · < i j ≤ n such that red ( σ i 1 . . . σ i j ) = τ . We let occr τ ( σ ) denote the number of τ occurrence in σ . We say σ avoids the pattern τ if τ does not occur in σ . Example π = 867932451 avoids pattern 132, contains pattern 123. occr 123 ( π ) = 2 since pattern occurrences are 6 , 7 , 9 and 3 , 4 , 5. τ is called a classical pattern. inversion − → pattern 21, coinversion − → pattern 12. Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 8 / 28

S n ( σ ) We let S n ( λ ) denote the set of permutations in S n avoiding λ . , the n th Catalan number. � = � = C n = � 2 n � � � � 1 � � S n (132) � S n (123) n +1 n C n is also the number of n × n Dyck paths. Let Λ = { λ 1 , . . . , λ r } , then S n (Λ) is the set of permutations in S n avoiding λ 1 , . . . , λ r . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 9 / 28

Our Problem Given two sets of permutations Λ = { λ 1 , . . . , λ r } and Γ = { γ 1 , . . . , γ s } , we study the distribution of classical patterns γ 1 , . . . , γ s in S n (Λ). Especially, we study pattern τ distribution in S n (132) and S n (123) in the case when τ is of length 3 and some special form. Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 10 / 28

Generating Function We define � Q Γ t n Q Γ Λ ( t , x 1 , . . . , x s ) = 1 + n , Λ ( x 1 , . . . , x s ) , n ≥ 1 where occr γ 1 ( σ ) · · · x occr γ s ( σ ) Q Γ � n , Λ ( x 1 , . . . , x s ) = x . s 1 σ ∈S n (Λ) Especially, we have Q γ � t n Q γ n ,λ ( x ) and Q γ � x occr γ ( σ ) . λ ( t , x ) = 1 + n ,λ ( x ) = n ≥ 1 σ ∈S n ( λ ) Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 11 / 28

Outline Motivation 1 Introduction 2 Wilf-equivalence of Q γ λ ( t , x ) 3 Recursions of Q γ λ ( t , x ) 4 Other Results and Open Problems 5 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 11 / 28

Wilf-equivalence Given a permutation σ , we denote the reverse of σ by σ r , the complement of σ by σ c , the reverse-complement of σ by σ rc , and the inverse of σ by σ − 1 . Example Let σ = 15324, then σ r = 42351 , σ c = 51342 , σ rc = 24315 , σ − 1 = 14352. Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 12 / 28

Wilf-equivalence S n (123) is closed under the operation reverse-complement. Both S n (123) and S n (132) are closed under the operation inverse. Thus, Theorem Given any permutation pattern γ , 123 ( t , x ) = Q γ rc 123 ( t , x ) = Q γ − 1 132 ( t , x ) = Q γ − 1 Q γ Q γ 123 ( t , x ) , 132 ( t , x ) . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 13 / 28

Wilf-equivalence When we let γ be a pattern of length 3, Corollary There are 4 Wilf-equivalent classes for S n (132) , (1) Q 123 132 ( t , x ) , (2) Q 213 132 ( t , x ) , (3) Q 231 132 ( t , x ) = Q 312 132 ( t , x ) , (4) Q 321 132 ( t , x ) , and there are 3 Wilf-equivalent classes for S n (123) , (1) Q 132 123 ( t , x ) = Q 213 123 ( t , x ) , (2) Q 231 123 ( t , x ) = Q 312 123 ( t , x ) , (3) Q 321 123 ( t , x ) . Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 14 / 28

Outline Motivation 1 Introduction 2 Wilf-equivalence of Q γ λ ( t , x ) 3 Recursions of Q γ λ ( t , x ) 4 Other Results and Open Problems 5 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 14 / 28

Method – Using Dyck Path Bijections We use Dyck path bijections to calculate the recursive formulas for Q γ λ ( t , x ). Krattenthaler Φ : S n (132) → D n , Elizalde and Deutsch Ψ : S n (123) → D n . 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 15 / 28

Method – Using Dyck Path Bijections Then, we the recursion of Dyck path by breaking the path at the first place it hits the diagonal to break it into 2 Dyck paths. Let D ( x ) be the generating function enumerating the number of Dyck paths of size n , D ( x ) = 1 + xD ( x ) 2 . D ( x ) x D ( x ) Recursion of Dyck path Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 16 / 28

Counting Length 2 pattern in S n (132) We first consider permutations that are avoiding 132 and the distribution of pattern of length 2, i.e. inv and coinv. We let Q n ( q ) = Q 12 � q coinv ( σ ) , n , 132 ( q ) = σ ∈S n (132) Q ( t , q ) = Q 12 � t n � q coinv ( σ ) , 132 ( t , q ) = 1 + n ≥ 1 σ ∈S n (132) � p inv ( σ ) q coinv ( σ ) . and P n ( p , q ) = σ ∈S n (132) Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 17 / 28

Counting Length 2 pattern in S n (132) We first consider permutations that are avoiding 132 and the distribution of pattern of length 2, i.e. inv and coinv. We let Q n ( q ) = Q 12 � q coinv ( σ ) , n , 132 ( q ) = σ ∈S n (132) Q ( t , q ) = Q 12 � t n � q coinv ( σ ) , 132 ( t , q ) = 1 + n ≥ 1 σ ∈S n (132) � p inv ( σ ) q coinv ( σ ) . and P n ( p , q ) = σ ∈S n (132) � n � Since inv ( σ ) + coinv ( σ ) = , we have the following relation about 2 P n ( p , q ) and Q n ( q ), � q � p ( n 2 ) − coinv ( σ ) q coinv ( σ ) = p ( n 2 ) Q n � P n ( p , q ) = . p σ ∈S n (132) Dun Qiu Pattern distribution in S n (132) and S n (123) July 11, 2018 17 / 28