Correlations in Pattern Avoidance Marisa Gaetz, Will Hardt, Shruthi - PowerPoint PPT Presentation

Correlations in Pattern Avoidance Marisa Gaetz, Will Hardt, Shruthi Sridhar, and Anh Quoc Tran UMN Twin Cities Combinatorics REU Problem 8 August 1, 2017 Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 1 / 24

Overview Preliminaries 1 Correlation Problem 2 Orignial Problem New Problem, u , v , w ∈ S 3 New Problem, v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 Characteristic Polynomial Problem 3 Avoiding k ( k − 1) ... 1 Avoiding t ( t − 1) .. 1 k ( k − 1) ... ( t + 1) Avoiding 12 ... k Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 2 / 24

Preliminaries A permutation is a bijection from { 1 , 2 , ..., n } to itself Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 3 / 24

Preliminaries A permutation is a bijection from { 1 , 2 , ..., n } to itself One-line notation Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 3 / 24

Preliminaries A permutation is a bijection from { 1 , 2 , ..., n } to itself One-line notation Definition A permutation π = π (1) π (2) ...π ( m ) contains a pattern σ = σ (1) σ (2) ...σ ( k ) if there exists a subsequence ( i 1 < ... < i k ) π ( i 1 ) π ( i 2 ) ...π ( i k ) of π with the same relative ordering as σ . Otherwise π avoids σ . Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 3 / 24

Preliminaries A permutation is a bijection from { 1 , 2 , ..., n } to itself One-line notation Definition A permutation π = π (1) π (2) ...π ( m ) contains a pattern σ = σ (1) σ (2) ...σ ( k ) if there exists a subsequence ( i 1 < ... < i k ) π ( i 1 ) π ( i 2 ) ...π ( i k ) of π with the same relative ordering as σ . Otherwise π avoids σ . Example ( 52 341 6 ) contains (213) , but avoids (132) S n ( σ 1 , ..., σ k ) = { π ∈ S n | π avoids σ 1 , ..., σ k } Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 3 / 24

The Original Problem Problem How do avoiding u and avoiding w correlate for random permutations in S n ? Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 4 / 24

The Original Problem Problem How do avoiding u and avoiding w correlate for random permutations in S n ? Solution (Joel Lewis) If u = (1 , 2 , ..., k ) , w = ( ℓ, ℓ − 1 , ..., 1) , then negative correlation (Erd˝ os-Szekeres). Otherwise, positive (Marcus-Tardos). Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 4 / 24

New Problem Problem How do avoiding u and avoiding w correlate for random permutations in S n ( v ) ? Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 5 / 24

New Problem Problem How do avoiding u and avoiding w correlate for random permutations in S n ( v ) ? Criterion Positive correlation if and only if (# S n ( v ))(# S n ( v , u , w )) > (# S n ( v , u ))(# S n ( v , w )) Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 5 / 24

Answer for u , v , w ∈ S 3 Case Simion and Schmidt (1985) give # S n (Π) for Π ⊂ S 3 , | Π | = 1 , 2 , 3 Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 6 / 24

Answer for u , v , w ∈ S 3 Case Simion and Schmidt (1985) give # S n (Π) for Π ⊂ S 3 , | Π | = 1 , 2 , 3 The following ( v , u , w ) triples negatively correlate: Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 6 / 24

Answer for u , v , w ∈ S 3 Case Simion and Schmidt (1985) give # S n (Π) for Π ⊂ S 3 , | Π | = 1 , 2 , 3 The following ( v , u , w ) triples negatively correlate: v ( u , w ) unordered pair (132) (123 , 231), (123 , 312), (213 , 231), (213 , 312), (231 , 312) (213) (123 , 231), (123 , 312), (132 , 231), (132 , 312), (231 , 312) (231) (132 , 213), (132 , 312), (132 , 321), (213 , 312), (213 , 321) (312) (132 , 213), (132 , 231), (132 , 321), (213 , 231), (213 , 321) Table: Complete list of ”interesting” negative correlations for u , v , w ∈ S 3 Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 6 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 Case Criterion Positive correlation if and only if (# S n ( k ... 1))(# S n ( w , ℓ... 1)) > (# S n ( ℓ... 1))(# S n ( w , k ... 1)) Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 7 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 Case Criterion Positive correlation if and only if (# S n ( k ... 1))(# S n ( w , ℓ... 1)) > (# S n ( ℓ... 1))(# S n ( w , k ... 1)) Theorem (Reifegerste) � n �� n � m − 1 � #( S n (132 , m ... 1)) = 1 n i i − 1 i =1 Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 7 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 cont. Theorem (Arriata/Regev) ( m − 1) 2 n #( S n ( m ... 1)) ∼ λ m n m ( m − 2) / 2 for some constant λ m Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 8 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 cont. Theorem (Arriata/Regev) ( m − 1) 2 n #( S n ( m ... 1)) ∼ λ m n m ( m − 2) / 2 for some constant λ m Conclusion w = 132 : positive correlation. Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 8 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 cont. Theorem (Arriata/Regev) ( m − 1) 2 n #( S n ( m ... 1)) ∼ λ m n m ( m − 2) / 2 for some constant λ m Conclusion w = 132 : positive correlation. Fact # S n (Π) = # S n (Π R ) = # S n (Π C ) i.e.: # S n (132) = # S n (213) and # S n (132 , m ... 1) = # S n (213 , m ... 1) Conclusion w = 213 : positive correlation. Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 8 / 24

v = ( k ... 1), u = ( ℓ... 1), w ∈ S 3 cont. Theorem (Arriata/Regev) ( m − 1) 2 n #( S n ( m ... 1)) ∼ λ m n m ( m − 2) / 2 for some constant λ m Conclusion w = 132 : positive correlation. Fact # S n (Π) = # S n (Π R ) = # S n (Π C ) i.e.: # S n (132) = # S n (213) and # S n (132 , m ... 1) = # S n (213 , m ... 1) Conclusion w = 213 : positive correlation. What about w = 231? Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 8 / 24

Main conjecture Albert, Atkinson and Vatter proved that any subclass of 231-avoiding permutations satisfies a linear recurrence. Conjecture For any 231-avoiding permutation π , T ( n ) = S n (231 , π ) satisfies a linear recurrence, and its characteristic polynomial has all positive real roots. This implies these coefficients form a P´ olya frequency sequence . Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 9 / 24

Avoiding 231 and k ( k − 1) ... 1 In the rest of the talk, we will denote D ( n , k ) = | S n (231 , k ( k − 1) ... 1) | . Note that D ( n , k ) = C n , the Catalan number, for n < k . Theorem � k � Let t = , then 2 � k − 1 � � k − 2 � D ( n , k ) = D ( n − 1 , k ) − D ( n − 2 , k ) + ... 1 2 � k − t � + ( − 1) t +1 D ( n − t , k ) t When k = 2 t, this result is true from n = t, and when k = 2 t + 1 , this is true from n = t + 1 . Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 10 / 24

Proof We proved the theorem by induction on k with the following recurrence Lemma � D ( n , k ) = D ( i , k ) D ( n − i − 1 , k − 1) 0 ≤ i < n combined with the identity Lemma � n − j � � n − 1 − j � ( − 1) j +1 =( − 1) j +1 + C j − 1 j j � n − 1 − i � j − 1 � ( − 1) i +1 C j − 1 − i − i i =1 Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 11 / 24

Another proof S n (231 , k ... 1) ↔ { Dyck paths of length 2 n and height ≤ k − 1 } f ( n ) = | S n (231 , k ... 1) | = number of directed paths connecting � u , � u + n � g . Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 12 / 24

Another proof The RHS: an acyclic weighted (all weighted 1) graph � N in a strip S , invariant under a shift � g . The LHS is � N ’s projection N on a cylinder O = S / Z � g Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 13 / 24

Another proof Galashin and Pylyavskyy proved a general (for any cylindrical network) version of the statement below: Theorem Denote f ( n ) the number of paths connecting � u , � u + n � g. Then for all but finitely many n, the sequence f satisfies a linear recurrence with characteristic polynomial d � ( − t ) d − r |C r ( N ) | Q N ( t ) = r =0 C r ( N ) is the set of r − tuples of disjoint simple cycles in N. Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 14 / 24

More conjecture Conjecture For any 231-avoiding pattern π , we can construct a cylindrical network � N such that f ( n ) = | S n (231 , π ) | has characteristic polynomial Q N ( t ) . Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 15 / 24

Avoiding 231 and k ( k − 1) ... 1 Proposition Let P k ( x ) denotes the characteristic polynomial for D ( n , k ) . Then P k has all real roots. We can prove that the roots of P k and P k +1 are interlaced by the following identities: P 2 k +1 ( x ) − P 2 k ( x ) = − P 2 k − 1 ( x ) P 2 k ( x ) − xP 2 k − 1 ( x ) = − P 2 k − 2 ( x ) Gaetz, Hardt, Sridhar, Tran (UMN TC) Correlations of Patterns August 1, 2017 16 / 24