Patterns and Statistics Bruce Sagan Department of Mathematics - PowerPoint PPT Presentation

Patterns and Statistics Bruce Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 sagan@math.msu.edu www.math.msu.edu/ ˜ sagan July 7, 2014

Generalities Permutation patterns and the major index maj-Wilf equivalence Other properties

Outline Generalities Permutation patterns and the major index maj-Wilf equivalence Other properties

The philosophy.

The philosophy. By combining the the theory of patterns with the theory of statistics, one opens up a whole realm of research problems waiting to be explored.

The method

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } .

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } . Let st : S n → { 0 , 1 , 2 , . . . } be a statistic on S n , n ≥ 0.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } . Let st : S n → { 0 , 1 , 2 , . . . } be a statistic on S n , n ≥ 0. Ex. 1. st = inv, the inversion number.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } . Let st : S n → { 0 , 1 , 2 , . . . } be a statistic on S n , n ≥ 0. Ex. 1. st = inv, the inversion number. 2. st = maj, the major index.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } . Let st : S n → { 0 , 1 , 2 , . . . } be a statistic on S n , n ≥ 0. Ex. 1. st = inv, the inversion number. 2. st = maj, the major index. 3. st = exc, the number of excedences.

The method Let S n , n ≥ 0, be a sequence of sets admitting a notion of pattern containment and avoidance. Ex. 1. S n = the n th symmetric group. 2. S n = all set partions of an n -element set. 3. S n = all words of length n over the positive integers. 4. S n = a relational structure on a set with n elements. Given t ∈ S k we let S n ( t ) = { s ∈ S n : s avoids t } . Let st : S n → { 0 , 1 , 2 , . . . } be a statistic on S n , n ≥ 0. Ex. 1. st = inv, the inversion number. 2. st = maj, the major index. 3. st = exc, the number of excedences. 4. st = lb, the left-bigger statistic.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t )

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | .

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | . Determine the st-Wilf equivalence classes.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | . Determine the st-Wilf equivalence classes. 2. The cardinalities | S n ( t ) | give interesting sequences such as Catalan numbers, Fibonacci numbers, and Schr¨ oder numbers.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | . Determine the st-Wilf equivalence classes. 2. The cardinalities | S n ( t ) | give interesting sequences such as Catalan numbers, Fibonacci numbers, and Schr¨ oder numbers. These sequences have many interesting properties such as recurrence relations, congruences, etc.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | . Determine the st-Wilf equivalence classes. 2. The cardinalities | S n ( t ) | give interesting sequences such as Catalan numbers, Fibonacci numbers, and Schr¨ oder numbers. These sequences have many interesting properties such as recurrence relations, congruences, etc. Find analogous properties of the ST n ( t ; q ) reducing to the old results for q = 1.

Study the generating functions � q st ( s ) . ST n ( t ) = ST n ( t ; q ) = s ∈ S n ( t ) Things to do. 1. Define t , u to be st-Wilf equivalent if ST n ( t ) = ST n ( u ) for all n ≥ 0. Note this implies ordinary Wilf equivalence since | S n ( t ) | = ST n ( t ; 1 ) = ST n ( u ; 1 ) = | S n ( u ) | . Determine the st-Wilf equivalence classes. 2. The cardinalities | S n ( t ) | give interesting sequences such as Catalan numbers, Fibonacci numbers, and Schr¨ oder numbers. These sequences have many interesting properties such as recurrence relations, congruences, etc. Find analogous properties of the ST n ( t ; q ) reducing to the old results for q = 1. 3. Study properties of the ST n ( t ; q ) which have no analogues when q = 1 such as degree, coefficients, unimodality, log concavity, real rootedness and so forth.

Papers with work in this area S n st Bach, Remmel des , lrm S n Barnabei, Bonetti, Elizalde, Silimbani S n maj Baxter maj , peak , valley S n Bloom S n maj Bousquet-M´ elou P n level , min , minmax Chan/Trongsiriwat S n inv Chen, Dai, Dokos, Dwyer, S Asc n asc , rlm Chen, Elizalde, Kasraoui, S S n inv , maj Dahlberg, S I n inv , maj Dahlberg, Dorward, Gerhard, Grubb, Π n Purcell, Reppuhn, S ls , lb , rs , rb Dokos, Dwyer, Johnson, S, Selsor inv , maj S n Duncan, Steingr´ ımsson Asc n asc , rlm Elizalde (also with Deutsch, Pak) des , exc , fp , S n Goyt (with Mathisen, S) Π n ls , rb Killpatrick ch , maj S n Kitaev, Remmel P n level , min Stanton, Simion Π n ls , lb , rs , rb

Outline Generalities Permutation patterns and the major index maj-Wilf equivalence Other properties

Let [ n ] = { 1 , 2 , . . . , n } , S n = { σ : σ is a permutaiton of [ n ] } , and S = ∪ n ≥ 0 S n .

Let [ n ] = { 1 , 2 , . . . , n } , S n = { σ : σ is a permutaiton of [ n ] } , and S = ∪ n ≥ 0 S n . Also, given π ∈ S k , let S n ( π ) = { σ ∈ S n : σ avoids π } .