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

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

Tandem walks A tandem-walk is a walk in Z 2 with step-set { N, W, SE } in the plane Z 2 in the half-plane { y ≥ 0 } in the quarter plane N 2 cf 2 queues in series SE N W y x

Link to Young tableaux of height ≤ 3 • There is a bijection between: tandem walks of length n staying in the quadrant N 2 , ending at ( i, j ) � Young tableaux of size n and height ≤ 3 , of shape j i 1 2 5 8 9 1 N 1 1 2 1 3 3 6 7 1 1 1 SE 0 1 3 5 6 2 9 1 0 3 end 4 1 2 W 4 7 8 1 tableau start walk (after k steps, current y = # N − # SE , current x = # SE − # W )

Link to Young tableaux of height ≤ 3 • There is a bijection between: tandem walks of length n staying in the quadrant N 2 , ending at ( i, j ) � Young tableaux of size n and height ≤ 3 , of shape j i m 1 2 5 8 9 1 N 1 1 2 1 3 3 6 7 1 1 1 SE 0 1 3 5 6 2 9 1 0 3 end 4 1 2 W 4 7 8 1 tableau start walk (after k steps, current y = # N − # SE , current x = # SE − # W ) • Let q [ n ; i, j ] := # tandem walks of length n in N 2 , ending at ( i, j ) Hook-length formula: for n of the form n = 3 m + 2 i + j we have ( i + 1)( j + 1)( i + j + 2) n q [ n ; i, j ] = m !( m + i + 1)!( m + i + j + 2)!

Algebraicity when the endpoint is free � q [ n ; i, j ] t n x i y j Let Q ( t ; x, y ) = n,i,j Theorem: [Gouyou-Beauchamps’89], [Bousquet-M´ elou,Mishna’10] Then Q ( t, 1 , 1) is the series counting Motzkin walks, Y = t · (1 + Y + Y 2 ) i.e., Y ≡ t Q ( t, 1 , 1) satisfies t · Y 2 = + + Y t · Y t

Bijection with Motzkin walks [Gouyou-Beauchamps’89] 1 2 1 2 5 8 9 1 1 1 3 1 1 6 2 5 9 1 0 3 6 7 1 0 1 3 3 4 4 1 2 7 8 1 Young tableau tandem walk in N 2 tandem walk in N 2 of height ≤ 3

Bijection with Motzkin walks [Gouyou-Beauchamps’89] 1 2 1 2 5 8 9 1 1 1 3 1 1 6 2 5 9 1 0 3 6 7 1 0 1 3 3 Robinson 4 4 1 2 7 Schensted 8 1 Young tableau tandem walk in N 2 tandem walk in N 2 involution of height ≤ 3 with no

Bijection with Motzkin walks [Gouyou-Beauchamps’89] 1 2 1 2 5 8 9 1 1 1 3 1 1 6 2 5 9 1 0 3 6 7 1 0 1 3 3 Robinson 4 4 1 2 7 Schensted 8 1 Young tableau tandem walk in N 2 tandem walk in N 2 involution of height ≤ 3 with no matching with no nesting

Bijection with Motzkin walks [Gouyou-Beauchamps’89] 1 2 1 2 5 8 9 1 1 1 3 1 1 6 2 5 9 1 0 3 6 7 1 0 1 3 3 Robinson 4 4 1 2 7 Schensted 8 1 Young tableau tandem walk in N 2 tandem walk in N 2 involution of height ≤ 3 with no no nesting FIFO Motzkin walk matching with no nesting

Bijection with Motzkin walks [Gouyou-Beauchamps’89] 1 2 1 2 5 8 9 1 1 1 3 1 1 6 2 5 9 1 0 3 6 7 1 0 1 3 3 Robinson 4 4 1 2 7 Schensted 8 1 Young tableau tandem walk in N 2 tandem walk in N 2 involution of height ≤ 3 with no no nesting FIFO Motzkin walk LIFO matching no crossing with no nesting

Reformulation with half-plane tandem-walks There is a bijection between: • tandem walks of length n staying in the quarter plane N 2 end start • tandem walks of length n � staying in the half-plane { y ≥ 0 } and ending at y = 0 y ⇔ start end t Rk: The bijection preserves the number of SE steps

An extension of the walk model y General model: level 3 step-set : • the SE step level 2 • every step ( − i, j ) (with i, j ≥ 0 ) level 1 level := i + j x SE Example: 1 0 1 9 2 start end 3 8 4 7 5 6

An extension of the walk model y General model: level 3 step-set : • the SE step level 2 • every step ( − i, j ) (with i, j ≥ 0 ) level 1 level := i + j x SE Example: 1 0 1 9 2 start end 3 8 4 7 5 6 There is still a bijection between: • general tandem walks of length n in the quarter plane N 2 • 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 bipolar orientation: (on planar maps) = acyclic orientation with a unique source S and a unique sink N with S, N incident to the outer face N inner vertex S inner face

Bipolar and marked bipolar orientations marked bipolar orientation: bipolar orientation: (on planar maps) a marked vertex A ′ � = N on left boundary = acyclic orientation a marked vertex A � = S on right boundary with a unique source S and a unique sink N with S, N incident to the outer face outdegree=1 N N inner vertex A A ′ S indegree=1 S inner face

The Kenyon et al. bijection The Kenyon et al. bijection [Kenyon, Miller, Sheffield, Wilson’16] bijection general tandem-walk (in Z 2 ) marked bipolar orientation SE step black vertex inner face of degree i + j +2 step ( − i, j ) N = A N = A N = A 1 0 N = A N = A N 1 9 2 ′ ′ ′ 3 A A ′ ′ A A A 3 4 5 1 2 8 A ′ = S A 4 7 5 6 S S S S S N N N N N A A A 6 7 1 0 9 8 ′ ′ A A A ′ ′ ′ A A A A S S S S S

The Kenyon et al. bijection [Kenyon, Miller, Sheffield, Wilson’16] • SE steps create a new black vertex N = A N = A N N A + SE-step + SE-step A A ′ A ′ A ′ A ′ S S S S • steps ( − i, j ) create a new inner face (of degree i + j + 2) N N N N A A ′ A ′ A ′ A A ′ A A S S S S

Parameter-correspondence in the bijection # “face-steps” # inner faces of level p of degree p + 2 # SE-steps # black vertices 1 + # steps # plain edges (not dashed) N δ ′ δ L ′ +1 minimal abscissa L A start A ′ L ′ end L +1 δ δ ′ S minimal ordinate

An involution on marked bipolar orientations N L ′ +1 δ A A ′ L +1 δ ′ S δ L ′ +1 L +1 δ ′

An involution on marked bipolar orientations N N δ ′ L ′ +1 δ L ′ +1 A A A ′ A ′ L +1 δ ↔ δ ′ L +1 δ ′ δ mirror S S δ δ ′ L ′ +1 L ′ +1 L +1 L +1 δ ′ δ

Effect of the involution on walks N N δ ′ δ L ′ +1 L ′ +1 A A involution A ′ A ′ L +1 L +1 δ ′ δ S S δ ↔ δ ′ minimal minimal δ ′ abscissa abscissa δ L L start start L ′ L ′ end end δ δ ′ minimal minimal ordinate ordinate

Quarter plane walks ↔ half-plane walks ending at y = 0 minimal minimal δ ′ abscissa abscissa δ L L δ ↔ δ ′ start start L ′ L ′ end end δ δ ′ minimal minimal ordinate ordinate • Specialize the involution at { L ′ = 0 , δ ′ = 0 } δ δ L L

Quarter plane walks ↔ half-plane walks ending at y = 0 minimal minimal δ ′ abscissa abscissa δ L L δ ↔ δ ′ start start L ′ L ′ end end δ δ ′ minimal minimal ordinate ordinate • Specialize the involution at { L ′ = 0 , δ ′ = 0 } δ δ L L • Specialize at { δ ′ ≤ a, L ′ ≤ b } ⇒ quarter plane walks starting at ( a, b )

Generating function expressions level 3 level 2 level 1 level 0 SE Let Q ( t ) be the generating function of general tandem-walks in N 2 • counted w.r.t. the length (variable t ) • with a weight z i for each “face-step” of level i Y = t · (1 + w 0 Y + w 1 Y 2 + w 2 Y 3 + · · · ) Then Y ≡ t Q ( t ) is given by where w i = z i + z i +1 + z i +2 + · · ·

Generating function expressions level 3 Y 4 level 2 Y 3 level 1 Y 2 level 0 Y SE Let Q ( t ) be the generating function of general tandem-walks in N 2 • counted w.r.t. the length (variable t ) • with a weight z i for each “face-step” of level i Y = t · (1 + w 0 Y + w 1 Y 2 + w 2 Y 3 + · · · ) Then Y ≡ t Q ( t ) is given by where w i = z i + z i +1 + z i +2 + · · ·

Generating function expressions level 3 Y 4 level 2 Y 3 level 1 Y 2 level 0 Y SE Let Q ( t ) be the generating function of general tandem-walks in N 2 • counted w.r.t. the length (variable t ) • with a weight z i for each “face-step” of level i Y = t · (1 + w 0 Y + w 1 Y 2 + w 2 Y 3 + · · · ) Then Y ≡ t Q ( t ) is given by where w i = z i + z i +1 + z i +2 + · · · Rk: alternative proof (earlier!) with obstinate kernel method

Generating function expressions level 3 Y 4 level 2 Y 3 level 1 Y 2 level 0 Y SE Let Q ( t ) be the generating function of general tandem-walks in N 2 • counted w.r.t. the length (variable t ) • with a weight z i for each “face-step” of level i Y = t · (1 + w 0 Y + w 1 Y 2 + w 2 Y 3 + · · · ) Then Y ≡ t Q ( t ) is given by where w i = z i + z i +1 + z i +2 + · · · Rk: alternative proof (earlier!) with obstinate kernel method Let Q ( a,b ) ( t ) := GF of general tandem walks in N 2 starting at ( a, b ) Rk: Then t Q ( a,b ) ( t ) = explicit polynomial in Y (with positive coefficients)

Quarter plane walks ending at ( i, 0) n q [ n ; i, 0] t n counts bipolar orientation of the form The series F i ( t ) := � N root-face degree i +2 root i S with t for # edges, and weight z r for each inner face of degree 0 ≤ r ≤ p