Orientations bipolaires et chemins tandem. Éric Fusy (CNRS/LIX) Travaux avec Mireille Bousquet-Mélou et Kilian Raschel

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1 Orientations bipolaires et chemins tandem Éric Fusy (CR/LIX) Travaux avec Mireille Bousquet-Mélou et Kilian Raschel Journées lea, 2017

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

3 Link to Young tableaux of height 3 There is a bijection between: tandem walks of length n staying in the quadrant 2, ending at (i, j) Young tableaux of size n and height 3, of shape j i start walk 11 end E W tableau (after k steps, current y = # #E, current x = #E #W )

4 Link to Young tableaux of height 3 There is a bijection between: tandem walks of length n staying in the quadrant 2, ending at (i, j) Young tableaux of size n and height 3, of shape j m i start walk 11 end E W tableau (after k steps, current y = # #E, current x = #E #W ) Let q[n; i, j] := # tandem walks of length n in 2, ending at (i, j) Hook-length formula: for n of the form n = 3m + 2i + j we have (i + 1)(j + 1)(i + j + 2)n q[n; i, j] = m!(m + i + 1)!(m + i + j + 2)!

5 lgebraicity when the endpoint is free Let Q(t; x, y) = n,i,j q[n; i, j]t n x i y j Theorem: [Gouyou-Beauchamps 89], [Bousquet-Mélou,Mishna 10] 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

6 ijection with Motzkin walks [Gouyou-Beauchamps 89] tandem walk in Young tableau of height 3

7 ijection with Motzkin walks [Gouyou-Beauchamps 89] tandem walk in Young tableau of height 3 Robinson chensted involution with no

8 ijection with Motzkin walks [Gouyou-Beauchamps 89] matching with no nesting tandem walk in Young tableau of height 3 Robinson chensted involution with no

9 ijection with Motzkin walks [Gouyou-Beauchamps 89] matching with no nesting tandem walk in Young tableau of height 3 Robinson chensted involution with no no nesting FIFO Motzkin walk

10 ijection with Motzkin walks [Gouyou-Beauchamps 89] tandem walk in Young tableau of height 3 Robinson chensted involution with no no nesting FIFO Motzkin walk LIFO no crossing matching with no nesting

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

12 n extension of the walk model General model: level 3 step-set: the E step every step ( i, j) (with i, j 0) level 2 level:= i + j level 1 Example: end start y E x

13 n extension of the walk model General model: level 3 step-set: the E step every step ( i, j) (with i, j 0) level 2 level:= i + j level 1 Example: end start y E x There is still a bijection between: general tandem walks of length n in the quarter plane 2 general tandem walks of length n in {y 0} ending at y = 0 The bijection preserves the number of E-steps and the number of steps in each level p 1

14 Bipolar and marked bipolar orientations bipolar orientation: (on planar maps) = acyclic orientation with a unique source and a unique sink with, incident to the outer face inner vertex inner face

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

16 The Kenyon et al. bijection general tandem-walk (in Z 2 ) E step step ( i, j) bijection [Kenyon, Miller, heffield, Wilson 16] marked bipolar orientation black vertex inner face of degree i+j+2 = = = = = =

17 The Kenyon et al. bijection E steps create a new black vertex [Kenyon, Miller, heffield, Wilson 16] = = + E-step +E-step steps ( i, j) create a new inner face (of degree i + j + 2)

18 Parameter-correspondence in the bijection # face-steps of level p # inner faces of degree p + 2 # E-steps # black vertices 1 + # steps # plain edges (not dashed) minimal abscissa start L L +1 L end minimal ordinate L+1

19 n involution on marked bipolar orientations L +1 L+1 L +1 L+1

20 n involution on marked bipolar orientations L +1 L +1 L+1 mirror L+1 L +1 L +1 L+1 L+1

21 Effect of the involution on walks L +1 L +1 involution minimal abscissa L+1 L+1 start L minimal abscissa L start L end minimal ordinate L end minimal ordinate

22 Quarter plane walks half-plane walks ending at y = 0 minimal abscissa start L minimal abscissa L start L end minimal ordinate L end minimal ordinate pecialize the involution at {L = 0, = 0} L L

23 Quarter plane walks half-plane walks ending at y = 0 minimal abscissa start L minimal abscissa L start L end minimal ordinate L end minimal ordinate pecialize the involution at {L = 0, = 0} L L pecialize at { a, L b} quarter plane walks starting at (a, b)

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

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

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

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

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

29 Quarter plane walks ending at (i, 0) The series F i (t) := n q[n; i, 0]tn counts bipolar orientation of the form root root-face degree i+2 i with t for # edges, and weight z r for each inner face of degree 0 r p symptotic enumeration q[n; i, 0] n C i γ n n 4 where C i = c α i (i + 1)(i + 2) from [Denisov-Wachtel 11] Rk: For undirected rooted maps M[n; i] n C i γ n n 5/2 where C i = c α i 4 i i ( ) 2i (with applications to peeling) i

30 Quarter plane walks ending at (i, 0) The series F i (t) := n q[n; i, 0]tn counts bipolar orientation of the form root root-face degree i+2 i with t for # edges, and weight z r for each inner face of degree 0 r p symptotic enumeration q[n; i, 0] n C i γ n n 4 where C i = c α i (i + 1)(i + 2) from [Denisov-Wachtel 11] Rk: For undirected rooted maps M[n; i] n C i γ n n 5/2 where C i = c α i 4 i i ( ) 2i (with applications to peeling) i Exact enumeration Let H i (t) = GF of Then F i (t) = H i (t) 1 t H i+2(t) + p r=0 (r + 1)z ih i+r+2 (t) Proof using obstinate kernel method or from Kenyon et al. + local operations i