Shuffling properties for products of random permutations Olivier - PowerPoint PPT Presentation

Shuffling properties for products of random permutations Olivier Bernardi (MIT) Joint work with Rosena Du, Alejandro Morales, Richard Stanley 3 9 7 10 2 1 8 5 11 4 6 Halifax, June 2012

Warm up Question: Let π be a uniformly random permutation of { 1 , 2 , . . . , n } . What is the probability that 1 , 2 , . . . , k are in distinct cycles of π ?

Warm up Question: Let π be a uniformly random permutation of { 1 , 2 , . . . , n } . What is the probability that 1 , 2 , . . . , k are in distinct cycles of π ? Answer: 1 k ! . Proof: Sampling process for π : build the cycles. 4 3 2 3 1 1 2 1 1 2

Warm up Let α = ( α 1 , α 2 , . . . , α k ) be a tuple of positive integers. Definition: A permutation π of { 1 , 2 , . . . , n } is said to be α -separated if letters from different blocks B 1 = { 1 , 2 , . . . , α 1 } , B 2 = { α 1 + 1 , α 1 + 2 , . . . , α 1 + α 2 } , . . . B k = { α 1 + · · · + α k − 1 + 1 , . . . , α 1 + · · · + α k } , are in different cycles of π .

Warm up Let α = ( α 1 , α 2 , . . . , α k ) be a tuple of positive integers. Definition: A permutation π of { 1 , 2 , . . . , n } is said to be α -separated if letters from different blocks B 1 = { 1 , 2 , . . . , α 1 } , B 2 = { α 1 + 1 , α 1 + 2 , . . . , α 1 + α 2 } , . . . B k = { α 1 + · · · + α k − 1 + 1 , . . . , α 1 + · · · + α k } , are in different cycles of π . Example: α = (2 , 2 , 1) . Blocks: { 1 , 2 } { 3 , 4 } { 5 } . 3 9 7 10 2 1 8 5 11 4 6

Warm up Let α = ( α 1 , α 2 , . . . , α k ) be a tuple of positive integers. Question: Let π be a uniformly random permutation of { 1 , 2 , . . . , n } . What is the probability that π is α -separated ? α 1 ! α 2 ! . . . α k ! Answer: ( α 1 + α 2 + . . . + α k )! .

Warm up Let α = ( α 1 , α 2 , . . . , α k ) be a tuple of positive integers. Question: Let π be a uniformly random permutation of { 1 , 2 , . . . , n } . What is the probability that π is α -separated ? α 1 ! α 2 ! . . . α k ! Answer: ( α 1 + α 2 + . . . + α k )! . Proof: Sampling process for π : build the cycles. 4 3 2 3 1 1 2 1 1 2

Results

Type of question Let λ, λ ′ be partitions of n . Let π, π ′ be uniformly random permutations of cycle-type λ, λ ′ . Let α be a composition of m ≤ n . What is the probability that the product π ◦ π ′ is α -separated?

Results Theorem [Du, Stanley]. Let π, π ′ be uniformly random n -cycles. The probability that 1 , 2 , . . . , k are in different cycles of product ππ ′ is 1 if n − k odd, k ! 1 2 k ! + otherwise. ( k − 2)!( n − k + 1)( n + k )

Results Theorem [Du, Stanley]. Let π, π ′ be uniformly random n -cycles. The probability that 1 , 2 , . . . , k are in different cycles of product ππ ′ is 1 if n − k odd, k ! 1 2 k ! + otherwise. ( k − 2)!( n − k + 1)( n + k ) + Extension to product ( n − j ) -cycle × n -cycle for k = 2 . Case k = 2 was conjectured by B´ ona.

More results: general composition α Theorem [BMDS]. Let π, π ′ be uniformly random n -cycles. Let α = ( α 1 , α 2 , . . . , α k ) be a composition of size m . The probability that the product ππ ′ is α -separated is ( − 1) n − m � n − 1 � m − k � ( n − m )! � k ( − 1) r � m − k �� n + r +1 � � i =1 α i ! k − 2 � r m + . � n + m � n + k + r � ( n + k )( n − 1)! � m − k r =0 r

More results: general composition α Theorem [BMDS]. Let π, π ′ be uniformly random n -cycles. Let α = ( α 1 , α 2 , . . . , α k ) be a composition of size m . The probability that the product ππ ′ is α -separated is ( − 1) n − m � n − 1 � m − k � ( n − m )! � k ( − 1) r � m − k �� n + r +1 � � i =1 α i ! k − 2 � r m + . � n + m � n + k + r � ( n + k )( n − 1)! � m − k r =0 r + Extension to π = uniformly random ( n − j ) -cycle.

More results: general composition α more cycles Theorem [BMDS]. Let π be a uniformly random permutation having p cycles. Let π ′ be a uniformly random n -cycle. Let α = ( α 1 , α 2 , . . . , α k ) be a composition of size m . The probability that the product ππ ′ is α -separated is �� n + k − 1 n − m ( n − m )! � k i =1 α i ! � 1 − k � c ( n − k − r + 1 , p ) � ( n − k − r + 1)! , c ( n, p ) n − m − r r r =0 where c ( n, p ) = [ x p ] x ( x + 1) · · · ( x + n − 1) “signless Stirling numbers of the first kind”.

More results: general composition α more cycles involutions Theorem [BMDS]. Let π be a uniformly random fixed-point free involution. Let π ′ be a uniformly random 2 N -cycle. Let α = ( α 1 , α 2 , . . . , α k ) be a composition of size m . The probability that the product ππ ′ is α -separated is � k i =1 α i ! (2 N − 1)!(2 N − 1)!! × �� 2 N + k − 1 min(2 N − m,N − k +1) � 2 k + r − N − 1 (2 N − k − r )! � 1 − k � . 2 N − m − r ( N − k − r + 1)! r r =0

More results: general composition α more cycles involutions symmetry Theorem [BMDS]. Let λ be a partition. Let π be a uniformly random permutation of type λ . Let π ′ be a uniformly random n -cycle. Let α = ( α 1 , α 2 , . . . , α k ) , β = ( β 1 , β 2 , . . . , β k ) be compositions of size m and length k . λ that the product ππ ′ is α -separated and λ and σ β The probabilities σ α β -separated are related by σ β σ α λ λ = . � k � k i =1 α i ! i =1 β i !

Strategy

Set up: Let α = ( α 1 , . . . , α k ) be a composition of m ≤ n . λ =proba that π ◦ π ′ is α -separated. Notation. σ α random permutation of type λ random n -cycle

Set up: Let α = ( α 1 , . . . , α k ) be a composition of m ≤ n . λ =proba that π ◦ π ′ is α -separated. Notation. σ α random permutation of type λ random n -cycle Def. For tuple A = ( A 1 , . . . , A k ) of disjoint subsets of { 1 , 2 , . . . , n } , we say that a permutation π is A -separated if elements in different blocks of A are in distinct cycles of π . Example: π is ( { 1 , 3 , 6 } , { 2 , 10 } ) -separated. 3 9 7 10 2 1 8 5 11 4 6

Set up: Let α = ( α 1 , . . . , α k ) be a composition of m ≤ n . λ =proba that π ◦ π ′ is α -separated. Notation. σ α random permutation of type λ random n -cycle Remark. σ α λ = proba that π ◦ (1 , 2 , . . . , n ) is A -separated. random permutation of type λ random subsets ( A 1 , . . . , A k ) with # A i = α i

Set up: Let α = ( α 1 , . . . , α k ) be a composition of m ≤ n . λ =proba that π ◦ π ′ is α -separated. Notation. σ α random permutation of type λ random n -cycle λ = proba that π − 1 ◦ (1 , 2 , . . . , n ) is A -separated. Remark. σ α random permutation of type λ random subsets ( A 1 , . . . , A k ) with # A i = α i

Set up: Let α = ( α 1 , . . . , α k ) be a composition of m ≤ n . λ =proba that π ◦ π ′ is α -separated. Notation. σ α random permutation of type λ random n -cycle λ = proba that π − 1 ◦ (1 , 2 , . . . , n ) is A -separated. Remark. σ α random permutation of type λ random subsets ( A 1 , . . . , A k ) with # A i = α i Lemma: # S α σ α λ λ = , n � � # C λ α 1 ,α 2 ,...,α k ,n − m where S α λ is set of triples ( A, π, ω ) such that • A = ( A 1 , . . . , A k ) is a tuple of disjoint subsets of [ n ] , with # A i = α i • permutation π has type λ , • permutation ω is A -separated, • π ◦ ω = (1 , 2 , . . . , n ) .

Set up: We want to count set S α λ of triples ( A, π, ω ) such that • A = ( A 1 , . . . , A k ) is a tuple of disjoint subsets of [ n ] , with # A i = α i • permutation π has type λ , • permutation ω is A -separated, • π ◦ ω = (1 , 2 , . . . , n ) . Example. Let n = 8 , α = (3 , 2) , λ = (3 , 3 , 2) . Triple ( A, π, ω ) is in S α λ where A = ( { 1 , 4 , 8 } , { 3 , 5 } ) , π = (1 , 7 , 4)(3 , 2 , 6)(5 , 8) , ω = (1 , 6)(2)(3 , 7 , 5)(4 , 8) .

A formula for colored factorization of long cycle Def. A colored permutation is a permutation with cycles colored in Z > 0 . It has color-type γ = ( γ 1 , . . . , γ k ) if there are exactly γ i elements of color i . Example: Colors 1 , 2 , 3 . 3 9 7 10 2 1 8 5 11 4 6 Color-type: γ = (4 , 3 , 4) .

A formula for colored factorization of long cycle Def. A colored permutation is a permutation with cycles colored in Z > 0 . It has color-type γ = ( γ 1 , . . . , γ k ) if there are exactly γ i elements of color i . Thm [Schaeffer Vassilieva 08, Vassilieva Morales 09]. Let γ, γ ′ be compositions of size n of length k, k ′ The number of colored permutations π, π ′ of color-types γ, γ ′ such that ππ ′ = (1 , 2 , . . . , n ) is B γ,γ ′ = n ( n − k )!( n − k ′ )! ( n − k − k ′ + 1)! .

Navigating between cycle-type and color-type Def. Symmetric functions in x = x 1 , x 2 , x 3 . . . Bases indexed by partitions λ = ( λ 1 , . . . , λ ℓ ) : p λ ( x ) = � ℓ i ≥ 1 x k i =1 p λ i ( x ) where p k ( x ) = � • Power basis: i . i ≥ 1 x γ i • Monomial basis: m λ ( x ) = � � i . γ 1 ,γ 2 ,... ∼ λ