An evolution of a permutation

Transcription

1 An evolution of a permutation Huseyin Acan April 28, 204 Joint work with Boris Pittel

2 Notation and Definitions S n is the set of permutations of {,..., n}

3 Notation and Definitions S n is the set of permutations of {,..., n} π = a a 2... a n

4 Notation and Definitions S n is the set of permutations of {,..., n} π = a a 2... a n (a i, a j ) is called an inversion if i < j and a i > a j

5 Notation and Definitions S n is the set of permutations of {,..., n} π = a a 2... a n (a i, a j ) is called an inversion if i < j and a i > a j π is called indecomposable (or connected) if there is no k < n such that {a,..., a k } = {,..., k} Otherwise it is decomposable is decomposable; is indecomposable

6 Notation and Definitions S n is the set of permutations of {,..., n} π = a a 2... a n (a i, a j ) is called an inversion if i < j and a i > a j π is called indecomposable (or connected) if there is no k < n such that {a,..., a k } = {,..., k} Otherwise it is decomposable is decomposable; is indecomposable C n = number of indecomposable permutations of length n (Sloane, sequence A00339) n C n = n! C k (n i)! k=

7 Problem σ(n, m) = permutation chosen u.a.r. from all permutations with n vertices and m inversions

8 Problem σ(n, m) = permutation chosen u.a.r. from all permutations with n vertices and m inversions Questions How does the connectedness probability of σ(n, m) change as m increases? Is there a (sharp) threshold for connectedness?

9 Problem σ(n, m) = permutation chosen u.a.r. from all permutations with n vertices and m inversions Questions How does the connectedness probability of σ(n, m) change as m increases? Is there a (sharp) threshold for connectedness? Definition T (n) is a sharp threshold for the property P if for any fixed ɛ > 0 m ( ɛ)t (n) = σ(n, m) does not have P whp m ( + ɛ)t (n) = σ(n, m) does have P whp

10 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions

11 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph

12 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

13 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

14 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

15 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

16 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

17 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

18 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

19 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

20 Permutation Graphs π = a a 2... a n G π (V, E) V = {, 2,..., n} E = set of inversions G π = permutation graph or inversion graph Example π =

21 Simple Facts π indecomposable G π connected

22 Simple Facts π indecomposable G π connected Vertex set of a connected component of G π consists of consecutive integers

23 Simple Facts π indecomposable G π connected Vertex set of a connected component of G π consists of consecutive integers (Comtet) If σ is chosen u.a.r. from S n, then Pr[σ is indecomposable] = 2/n + O(/n 2 )

24 Connectivity and descent sets Connectivity set of π C(π) = {i [n ] : a j < a k for all j i < k} C( ) = {5}

25 Connectivity and descent sets Connectivity set of π C(π) = {i [n ] : a j < a k for all j i < k} C( ) = {5} Descent set of π D(π) = {i [n ] : a i > a i+ } D( ) = {2, 7}

26 Connectivity and descent sets Connectivity set of π C(π) = {i [n ] : a j < a k for all j i < k} C( ) = {5} Descent set of π D(π) = {i [n ] : a i > a i+ } D( ) = {2, 7} Proposition (Stanley) Given I [n ], {ω S n : I C(ω)} {ω S n : I D(ω)} = n!

27 Permutations with given number of cycles π(n, m) = permutation chosen u.a.r from all permutations of {,..., n} with m cycles p(n, m) = Pr[π(n, m) is connected] Theorem (R. Cori, C. Matthieu, and J.M. Robson - 202) (i) p(n, m) is decreasing in m (ii) p(n, m) f (c) as n and m/n c

28 Erdős-Rényi Graphs G(n, m) : Uniform over all graphs on [n] with exactly m edges Connectedness probability of G(n, m) increases with m Sharp threshold: n log n/2

29 Erdős-Rényi Graphs G(n, m) : Uniform over all graphs on [n] with exactly m edges Connectedness probability of G(n, m) increases with m Sharp threshold: n log n/2 Graph Process G n Start with n isolated vertices Add an edge chosen u.a.r. at each step G(n, m) is the snapshot at the m-th step of the process G(n, m) G(n, m + )

30 Erdős-Rényi Graph G(n, m) 0 n k 2 k ( ) n/2 ( ) n log n/2 ( ) n 4/3 ( ) ( n 2) n (k 2)/(k ) : components of size k n/2: giant component n log n/2: connectedness n 4/3 : 4-clique

31 Question: Is there a similar process for σ(n, m) (or G σ(n,m) ) such that. Uniform distribution is achieved after each step 2. Existing inversions (edges of G σ(n,m) ) are preserved

32 Question: Is there a similar process for σ(n, m) (or G σ(n,m) ) such that. Uniform distribution is achieved after each step 2. Existing inversions (edges of G σ(n,m) ) are preserved Answer: NO

33 Evolution of a Permutation: Model Swap neighbors if they are in the correct order

34 Evolution of a Permutation: Model Swap neighbors if they are in the correct order Example (n=4) 234 /3 /3 /

35 Evolution of a Permutation: Model Swap neighbors if they are in the correct order Example (n=4) 234 /3 /3 / ??

36 Evolution of a Permutation: Model Swap neighbors if they are in the correct order Example (n=4) 234 /3 /3 / ?? Preserves the existing inversions (edges in the permutation) No uniformity

37 Question: Is there a process for G σ(n,m) (or σ(n, m)) such that. Uniform distribution is achieved after each step 2. Once the graph (permutation) becomes connected, it is connected always

38 Question: Is there a process for G σ(n,m) (or σ(n, m)) such that. Uniform distribution is achieved after each step 2. Once the graph (permutation) becomes connected, it is connected always Answer: YES

39 Inversion Sequences Inversion sequence of π = a a 2... a n is (x,..., x n ) x j = #{i : i < j and a i > a j } 0 x j j permutations of [n] (x,..., x n ) where 0 x i i Example (x, x 2, x 3, x 4, x 5 ) = (0,, 0, 3, 3) π = 4, 3, 5,, 2

40 Evolution of a Permutation: Model 2 Increase one of the components in the inversion sequence by Not all the inversions are protected Once the permutation becomes connected, it continues to be connected Example (n=4)

41 Evolution of a Permutation: Model 2 Increase one of the components in the inversion sequence by Not all the inversions are protected Once the permutation becomes connected, it continues to be connected Example (n=4) 0000

42 Evolution of a Permutation: Model 2 Increase one of the components in the inversion sequence by Not all the inversions are protected Once the permutation becomes connected, it continues to be connected Example (n=4)

43 Evolution of a Permutation: Model 2 Increase one of the components in the inversion sequence by Not all the inversions are protected Once the permutation becomes connected, it continues to be connected Example (n=4)

44 Inv. Sequence Permutation Graph

45 Inv. Sequence Permutation Graph

46 Inv. Sequence Permutation Graph

47 Inv. Sequence Permutation Graph

48 Inv. Sequence Permutation Graph

49 Inv. Sequence Permutation Graph

50 Inv. Sequence Permutation Graph

51 Inv. Sequence Permutation Graph

52 f (n, k) = number of permutations of [n] with k inversions. number of integer solutions of x + + x n = k, 0 x i i

53 f (n, k) = number of permutations of [n] with k inversions. number of integer solutions of x + + x n = k, 0 x i i 2. k balls are placed into n boxes box i has capacity i

54 f (n, k) = number of permutations of [n] with k inversions. number of integer solutions of x + + x n = k, 0 x i i 2. k balls are placed into n boxes box i has capacity i n f (n, k) = [z k ] ( + z + + z j ) j=0 n = [z k ]( z) n ( z j ) j=

55 The Process Start with (0, 0,..., 0) Each time increase exactly one of the components by X(k) = (X (k),..., X n (k)) after step k is uniformly distributed

56 The Process Start with (0, 0,..., 0) Each time increase exactly one of the components by X(k) = (X (k),..., X n (k)) after step k is uniformly distributed Example (0, 0, 0, 0) (0, 0,, 0) (0, 0,, ) (0, 0, 2, ) (0, 0, 2, 2) (0,, 2, 2) (0,, 2, 3)

57 Goal: Finding p(x(k)), a (conditional) probability distribution for the (k + )st addition OR

58 Goal: Finding p(x(k)), a (conditional) probability distribution for the (k + )st addition OR Transition matrix ρ n,k f (n, k) f (n, k + ) matrix rows are indexed by inversion sequences with sum k columns are indexed by inversion sequences with sum k +

59 Example (n=3) f (3, 0) =, f (3, ) = 2, s(3, 2) = 2, and s(3, 3) =. ρ 3,0 = ρ 3, = ρ 3,2 = 00 [ 00 ] 000 /2 /2 0 [ 002 ] [ ] 000 /2 /

60 Theorem Transition matrices exist for all n and for all possible values of m.

61 Theorem Transition matrices exist for all n and for all possible values of m. Sketch Proof Induction on n Order the sequences with reverse lexicographic order y n = 0 y n = y n = 2... y n = n 2 y n = n x n = 0 ρ n,m β I x n = ρ n,m β 2 I x n = 2. ρ n,m x n = n 2 ρ n,m n+2 β n I x n = n ρ n,m n+ ρ (n, m j) = ( β j+ )ρ n,m j Find constants β,..., β n such that all the column sums are equal to f (n, m)/f (n, m + )

62 β β β 0 β β 2 0 β β 2 0 β column sums must be 5/6 β 3 2 β 3 2 β 3

63 β β β 0 β β 2 0 β β 2 0 β column sums must be 5/6 β 3 2 β 3 2 β /2 7/ /2 0 7/ /2 0 9/ /2 0 9/ /2 /2 0/2

64 Definition An index t (t ) is a decomposition point if (X t+,..., X n ) is an inversion sequence, i.e., if X t+ 0, X t+2,... X n n t

65 Definition An index t (t ) is a decomposition point if (X t+,..., X n ) is an inversion sequence, i.e., if X t+ 0, X t+2,... X n n t number of components = number of decomposition points +

66 Definition An index t (t ) is a decomposition point if (X t+,..., X n ) is an inversion sequence, i.e., if X t+ 0, X t+2,... X n n t number of components = number of decomposition points + Corollary Pr[σ(n, m) is indecomposable] is non-decreasing in m

67 C(σ) := number of components in G σ(n,m) Theorem If (i) m = 6n π 2 [ log(n) log log(n) + log(2/π) 2/π 2 + x n ] (ii) x n = o(log log log n) then d TV [C(σ), Poisson(e xn )] (log n) +ɛ for any ɛ > 0.

68 C(σ) := number of components in G σ(n,m) Theorem If (i) m = 6n π 2 [ log(n) log log(n) + log(2/π) 2/π 2 + x n ] (ii) x n = o(log log log n) then d TV [C(σ), Poisson(e xn )] (log n) +ɛ for any ɛ > 0. Remarks. If x n c, then C(σ) d Poisson(e c ) 2. T (n) = 6n π 2 [log n log log n] is a sharp threshold for connectedness of G σ(n,m)

69 Idea of the Proof for x n c. Need D n, the number of decomposition points

70 Idea of the Proof for x n c. Need D n, the number of decomposition points ν = 2m log n/n Mark t if (X t+,..., X t+ν ) is an inversion sequence M n = number of marked points

71 Idea of the Proof for x n c. Need D n, the number of decomposition points ν = 2m log n/n Mark t if (X t+,..., X t+ν ) is an inversion sequence M n = number of marked points 2. Whp M n = D n as n 3. Pr[t is marked] e c /n [ (Mn ) ] 4. E k = E (e c ) k k 5. M n Poisson(e c ) in distribution k!

72 L min = size of the smallest component L max = size of the largest block (component) Theorem If m = 6n π 2 [ log(n) log log(n) + log(2/π) 2/π 2 x n ] x n = o(log log log n) and x n then. lim n Pr[L min ne 2xn y] = e y, for any constant y 0 2. lim n P[L max ne xn (x n + z)] = e e z, for constant z 0 Note: Expected number of decomposition points e xn

73 Remark Divide the interval [0, ] into k intervals with k randomly chosen points. L min, L max = smallest and largest intervals, respectively Pr[L min y/k 2 ] e y as k Pr[L max log k+z k ] e e z as k

74 Question: Conditioned on {the number of blocks in σ(n, m) = k}, do we have (L /n,..., L k /n) (η,..., η k ) as n where L j = size of the j th block in σ(n, m) η j = size of the j th interval in [0, ]?

75 Chord Diagrams and Intersection Graphs Chord Diagram matching of 2n points Intersection Graph V = chords, E = crossings (,9) (5,) 2 7 (2,2) (6,7) (3,8) (4,0) Number of chord diagrams: (2n )!! = (2n )(2n 3) (3) ()

76 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

77 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

78 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

79 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

80 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

81 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

82 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

83 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle

84 Permutations as Chord Diagrams Relabel the points on the lower semicircle Draw the chords from the upper semicircle to the lower semicircle Permutation= 25436

85 pointed hypermaps indecomposable permutations Definition A labeled pointed hypermap on [n] is a triple (σ, θ, r) S n S n [n] such that < σ, θ > acts transitively on [n]. Example m σ = (abel)(cdk)(fgi)(hjm) h g i θ = (adf )(bjc)(egh)(ilkm) r = m j e b f a l d k c

86 THANK YOU!