Orientations bipolaires et chemins tandem Eric Fusy (CNRS/LIX) - - PowerPoint PPT Presentation

Orientations bipolaires et chemins tandem

Journ´ ees Alea, 2017 ´ Eric Fusy (CNRS/LIX) Travaux avec Mireille Bousquet-M´ elou et Kilian Raschel

Tandem walks

A tandem-walk is a walk in Z2 with step-set {N, W, SE}

in the plane Z2 in the half-plane {y ≥ 0} in the quarter plane N2 cf 2 queues in series y x SE N W

Link to Young tableaux of height ≤ 3

• There is a bijection between:

tandem walks of length n staying in the quadrant N2, ending at (i, j) Young tableaux of size n and height ≤ 3, of shape

• i

j 1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

walk tableau N SE W (after k steps, current y = #N − #SE, current x = #SE − #W) start end

Link to Young tableaux of height ≤ 3

• There is a bijection between:

tandem walks of length n staying in the quadrant N2, ending at (i, j) Young tableaux of size n and height ≤ 3, of shape

• i

j 1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

walk tableau N SE W (after k steps, current y = #N − #SE, current x = #SE − #W) start end

• Let q[n; i, j] := # tandem walks of length n in N2, ending at (i, j)

m Hook-length formula: for n of the form n = 3m + 2i + j we have q[n; i, j] = (i + 1)(j + 1)(i + j + 2)n m!(m + i + 1)!(m + i + j + 2)!

Algebraicity when the endpoint is free

Let Q(t; x, y) =

• n,i,j

q[n; i, j]tnxiyj Then Q(t, 1, 1) is the series counting Motzkin walks, i.e., Y ≡ t Q(t, 1, 1) satisfies Y = t · (1 + Y + Y 2) Y = t + t · Y + t · Y 2 Theorem: [Gouyou-Beauchamps’89], [Bousquet-M´

elou,Mishna’10]

Bijection with Motzkin walks

1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

[Gouyou-Beauchamps’89]

tandem walk in N2 tandem walk in N2 Young tableau

• f height ≤ 3

Bijection with Motzkin walks

1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

[Gouyou-Beauchamps’89]

Robinson Schensted involution with no tandem walk in N2 tandem walk in N2 Young tableau

• f height ≤ 3

Bijection with Motzkin walks

1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

[Gouyou-Beauchamps’89]

Robinson Schensted involution with no matching with no nesting tandem walk in N2 tandem walk in N2 Young tableau

• f height ≤ 3

Bijection with Motzkin walks

1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

[Gouyou-Beauchamps’89]

Robinson Schensted involution with no matching with no nesting Motzkin walk no nesting FIFO tandem walk in N2 tandem walk in N2 Young tableau

• f height ≤ 3

Bijection with Motzkin walks

1 2 5 8 9 1 1 3 6 7 1 0 1 3 4 1 2

1 2 3 4 5 6 7 8 9 1 1 1 1 2 1 3

[Gouyou-Beauchamps’89]

Robinson Schensted involution with no matching with no nesting Motzkin walk no nesting no crossing FIFO LIFO tandem walk in N2 tandem walk in N2 Young tableau

• f height ≤ 3

Reformulation with half-plane tandem-walks

There is a bijection between:

• tandem walks of length n
• tandem walks of length n

staying in the quarter plane N2 staying in the half-plane {y ≥ 0} and ending at y = 0

• ⇔

start end start end Rk: The bijection preserves the number of SE steps t y

An extension of the walk model

General model: step-set: • the SE step level:= i + j level 1 level 2 level 3 SE x y

• every step (−i, j) (with i, j ≥ 0)

1 2 3 4 5 6 7 8 9 1

start end Example:

An extension of the walk model

General model: step-set: • the SE step level:= i + j level 1 level 2 level 3 SE x y

• every step (−i, j) (with i, j ≥ 0)

1 2 3 4 5 6 7 8 9 1

start end Example: There is still a bijection between:

• general tandem walks of length n in the quarter plane N2
• general tandem walks of length n in {y ≥ 0} ending at y = 0

The bijection preserves the number of SE-steps and the number of steps in each level p ≥ 1

Bipolar and marked bipolar orientations

N S bipolar orientation: inner vertex inner face

(on planar maps) = acyclic orientation with a unique source S and a unique sink N with S, N incident to the outer face

Bipolar and marked bipolar orientations

A N S A′ N S bipolar orientation: marked bipolar orientation: indegree=1

• utdegree=1

inner vertex inner face

a marked vertex A′ =N on left boundary a marked vertex A=S on right boundary (on planar maps) = acyclic orientation with a unique source S and a unique sink N with S, N incident to the outer face

The Kenyon et al. bijection

N=A A

′=S

S N A A

′

S A

′

S A

′

S A

′

S N A A

′

N A

′

A S A S N A

′

S A

′

A N S A

′

A N

1 2 3 4 5 6 7 8 9 1

1 3 4 5 6 7 8 9 1 S N =A A

′

2 N =A N =A N =A

general tandem-walk (in Z2)

SE step black vertex

The Kenyon et al. bijection

step (−i, j) inner face of degree i+j+2

marked bipolar orientation

bijection

[Kenyon, Miller, Sheffield, Wilson’16]

The Kenyon et al. bijection

N S A′ A N S A′ A N S A′ N S A′ A A

• steps (−i, j) create a new inner face (of degree i + j + 2)
• SE steps create a new black vertex

N S A′ A N S A′ A N =A S A′ S A′ N =A

+ SE-step + SE-step

[Kenyon, Miller, Sheffield, Wilson’16]

Parameter-correspondence in the bijection

1 + # steps # plain edges (not dashed) # “face-steps” # inner faces # SE-steps # black vertices

• f level p
• f degree p + 2

end

δ L

minimal

L′

A S N

L′+1 δ L+1

• rdinate

δ′

minimal abscissa

δ′

A′ start

An involution on marked bipolar orientations

L′+1 δ′ δ A N S A′ L+1 δ L′+1 L+1 δ′

An involution on marked bipolar orientations

L′+1 δ′ δ A N S A′ L+1 δ′ δ L′+1 L+1 A S N A′ δ L′+1 L+1 δ′

mirror

L′+1 L+1 δ δ′ δ ↔ δ′

Effect of the involution on walks

A S N

L′+1 δ L+1 δ′

A′

L+1 δ′ L′+1 δ

end

δ L

minimal

L′

• rdinate

δ′

minimal abscissa start N S A′ A end

L

minimal

L′

• rdinate

δ′

minimal abscissa start

δ

involution δ ↔ δ′

Quarter plane walks ↔ half-plane walks ending at y = 0

end

δ L

minimal

L′

• rdinate

δ′

minimal abscissa start end

L

minimal

L′

• rdinate

δ′

minimal abscissa start

δ

• Specialize the involution at {L′ = 0, δ′ = 0}

L δ δ L

δ ↔ δ′

Quarter plane walks ↔ half-plane walks ending at y = 0

end

δ L

minimal

L′

• rdinate

δ′

minimal abscissa start end

L

minimal

L′

• rdinate

δ′

minimal abscissa start

δ

• Specialize the involution at {L′ = 0, δ′ = 0}

L δ δ L

• Specialize at {δ′ ≤ a, L′ ≤ b} ⇒ quarter plane walks starting at (a, b)

δ ↔ δ′

Generating function expressions

level 1 level 2 level 3 SE Let Q(t) be the generating function of general tandem-walks in N2

• counted w.r.t. the length (variable t)
• with a weight zi for each “face-step” of level i

Then Y ≡ t Q(t) is given by Y = t · (1 + w0Y + w1Y 2 + w2Y 3 + · · · ) where wi = zi + zi+1 + zi+2 + · · · level 0

Generating function expressions

level 1 level 2 level 3 SE Let Q(t) be the generating function of general tandem-walks in N2

• counted w.r.t. the length (variable t)
• with a weight zi for each “face-step” of level i

Then Y ≡ t Q(t) is given by Y = t · (1 + w0Y + w1Y 2 + w2Y 3 + · · · ) where wi = zi + zi+1 + zi+2 + · · · level 0 Y Y 2 Y 3 Y 4

Generating function expressions

level 1 level 2 level 3 SE Let Q(t) be the generating function of general tandem-walks in N2

• counted w.r.t. the length (variable t)
• with a weight zi for each “face-step” of level i

Then Y ≡ t Q(t) is given by Y = t · (1 + w0Y + w1Y 2 + w2Y 3 + · · · ) where wi = zi + zi+1 + zi+2 + · · · level 0 Y Y 2 Y 3 Y 4 Rk: alternative proof (earlier!) with obstinate kernel method

Generating function expressions

level 1 level 2 level 3 SE Let Q(t) be the generating function of general tandem-walks in N2

• counted w.r.t. the length (variable t)
• with a weight zi for each “face-step” of level i

Then Y ≡ t Q(t) is given by Y = t · (1 + w0Y + w1Y 2 + w2Y 3 + · · · ) where wi = zi + zi+1 + zi+2 + · · · level 0 Y Y 2 Y 3 Y 4 Let Q(a,b)(t) := GF of general tandem walks in N2 starting at (a, b) Then t Q(a,b)(t) = explicit polynomial in Y (with positive coefficients) Rk: Rk: alternative proof (earlier!) with obstinate kernel method

Quarter plane walks ending at (i, 0)

The series Fi(t) :=

n q[n; i, 0]tn counts bipolar orientation of the form

with t for # edges, and weight zr for each inner face of degree 0 ≤ r ≤ p i root S N root-face degree i+2

Quarter plane walks ending at (i, 0)

The series Fi(t) :=

n q[n; i, 0]tn counts bipolar orientation of the form

with t for # edges, and weight zr for each inner face of degree 0 ≤ r ≤ p i root S N root-face degree i+2 Asymptotic enumeration q[n; i, 0] ∼n→∞ Ci · γn · n−4 where Ci = c · αi(i + 1)(i + 2) Rk: For undirected rooted maps M[n; i] ∼n→∞ Ci · γn · n−5/2

from [Denisov-Wachtel’11]

where Ci = c · αi4−ii 2i

i

• (with applications to peeling)

Quarter plane walks ending at (i, 0)

The series Fi(t) :=

n q[n; i, 0]tn counts bipolar orientation of the form

with t for # edges, and weight zr for each inner face of degree 0 ≤ r ≤ p i root S N root-face degree i+2 Asymptotic enumeration q[n; i, 0] ∼n→∞ Ci · γn · n−4 where Ci = c · αi(i + 1)(i + 2) Rk: For undirected rooted maps M[n; i] ∼n→∞ Ci · γn · n−5/2

from [Denisov-Wachtel’11]

where Ci = c · αi4−ii 2i

i

• (with applications to peeling)

Exact enumeration

Let Hi(t) = GF of

i

Then Fi(t) = Hi(t) − 1

t Hi+2(t)

+ p

r=0(r + 1)ziHi+r+2(t)

Proof using obstinate kernel method

• r from Kenyon et al. + local operations