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

Permutation 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 joint work with T. Dokos (Ohio State), T. Dwyer (U. Florida), B. Johnson (Michigan State), and K. Selsor (U. South Carolina) May 9, 2012

Pattern containment and avoidance Permutation statistics: inversions Permutation statistics: major index q -Catalan numbers Multiple restrictions Future work

Outline Pattern containment and avoidance Permutation statistics: inversions Permutation statistics: major index q -Catalan numbers Multiple restrictions Future work

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic.

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485.

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485. We say σ avoids π if σ does not contain π and let Av n ( π ) = { σ ∈ S n : σ avoids π } .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485. We say σ avoids π if σ does not contain π and let Av n ( π ) = { σ ∈ S n : σ avoids π } . Ex. If π ∈ S k then Av k ( π )

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485. We say σ avoids π if σ does not contain π and let Av n ( π ) = { σ ∈ S n : σ avoids π } . Ex. If π ∈ S k then Av k ( π ) = S k − { π } .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485. We say σ avoids π if σ does not contain π and let Av n ( π ) = { σ ∈ S n : σ avoids π } . Ex. If π ∈ S k then Av k ( π ) = S k − { π } . Say that π and π ′ are Wilf equivalent , π ≡ π ′ , if for all n ≥ 0 # Av n ( π ) = # Av n ( π ′ ) .

Two sequences of distinct integers π = a 1 a 2 . . . a k and σ = b 1 b 2 . . . b k are order isomorphic if, for all i and j , a i < a j ⇐ ⇒ b i < b j . Ex. The sequences π = 132 and σ = 485 are order isomorphic. Let S n be the symmetric group of all permutations of { 1 , . . . , n } and let S = ∪ n ≥ 0 S n . If π, σ ∈ S then σ contains π as a pattern if there is a subsequence σ ′ of σ order isomorphic to π . Ex. σ = 42183756 contains π = 132 because of σ ′ = 485. We say σ avoids π if σ does not contain π and let Av n ( π ) = { σ ∈ S n : σ avoids π } . Ex. If π ∈ S k then Av k ( π ) = S k − { π } . Say that π and π ′ are Wilf equivalent , π ≡ π ′ , if for all n ≥ 0 # Av n ( π ) = # Av n ( π ′ ) . Theorem For any π ∈ S 3 we have # Av n ( π ) = C n , the nth Catalan number.

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 .

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. 132 =

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. 132 = The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m .

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. R 90 ( 132 ) = 132 = = 231 The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m .

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. R 90 ( 132 ) = 132 = = 231 The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m . Note that for any ρ ∈ D 4 : σ contains π ⇐ ⇒ ρ ( σ ) contains ρ ( π ) ,

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. R 90 ( 132 ) = 132 = = 231 The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m . Note that for any ρ ∈ D 4 : σ contains π ⇐ ⇒ ρ ( σ ) contains ρ ( π ) , ∴ σ avoids π ⇐ ⇒ ρ ( σ ) avoids ρ ( π ) ,

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. R 90 ( 132 ) = 132 = = 231 The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m . Note that for any ρ ∈ D 4 : σ contains π ⇐ ⇒ ρ ( σ ) contains ρ ( π ) , ∴ σ avoids π ⇐ ⇒ ρ ( σ ) avoids ρ ( π ) , ρ ( π ) ≡ π. ∴

The diagram of π = a 1 . . . a n is ( 1 , a 1 ) , . . . , ( n , a n ) ∈ Z 2 . Ex. R 90 ( 132 ) = 132 = = 231 The dihedral group D 4 of symmetries of the square acts on S n : D 4 = { R 0 , R 90 , R 180 , R 270 , r 0 , r 1 , r − 1 , r ∞ } where R θ is rotation counter-clockwise through θ degrees and r m is reflection in a line of slope m . Note that for any ρ ∈ D 4 : σ contains π ⇐ ⇒ ρ ( σ ) contains ρ ( π ) , ∴ σ avoids π ⇐ ⇒ ρ ( σ ) avoids ρ ( π ) , ρ ( π ) ≡ π. ∴ These Wilf equivalences are called trivial .

Outline Pattern containment and avoidance Permutation statistics: inversions Permutation statistics: major index q -Catalan numbers Multiple restrictions Future work

A permutation statistic is st : S → { 0 , 1 , 2 , . . . } .

A permutation statistic is st : S → { 0 , 1 , 2 , . . . } . The inversion number of π = a 1 . . . a n is inv π = # { ( i , j ) : i < j and a i > a j } .

A permutation statistic is st : S → { 0 , 1 , 2 , . . . } . The inversion number of π = a 1 . . . a n is inv π = # { ( i , j ) : i < j and a i > a j } . Ex. If π = 24135 then inv π = # { ( 1 , 3 ) , ( 2 , 3 ) , ( 2 , 4 ) } = 3.