unified construction between blossoming trees and...

Unified construction betweenblossoming trees and planar maps

Shanghai Jiao Tong University, November 23rd 2013

Marie Albenque (CNRS, Paris)Joint work with Dominique Poulalhon

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.

plane map = plane graph + cyclic order of neigbours around each vertex.

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.

Faces = connected components of the plane when the edge are removed

plane map = plane graph + cyclic order of neigbours around each vertex.

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.

Plane maps are rooted.

Faces = connected components of the plane when the edge are removed

plane map = plane graph + cyclic order of neigbours around each vertex.

Plane maps are rooted.

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.

Faces = connected components of the plane when the edge are removed

There is one special face which is infinite: the outer face.

plane map = plane graph + cyclic order of neigbours around each vertex.

Some families of maps

Simple triangulationsfaces are trianglesno loops nor multiple edges

Some families of maps

Simple triangulationsfaces are trianglesno loops nor multiple edges

4-regular mapsall vertices have degree 4

Some families of maps

Simple triangulationsfaces are trianglesno loops nor multiple edges

4-regular mapsall vertices have degree 4

Bipolar orientationsacyclic orientationonly one sink tonly one source s

s

t

Why maps ?

What the motivation for studying maps instead of graphs ?

Because maps have more structure than graphs,they are actually simpler to study.

Why maps ?

What the motivation for studying maps instead of graphs ?

Because maps have more structure than graphs,they are actually simpler to study.

Euler Formula : # vertices+#faces = 2+#edges

A triangulation with 2n faces has 3n edges and n+ 2 vertices.

Why maps ?

What the motivation for studying maps instead of graphs ?

Because maps have more structure than graphs,they are actually simpler to study.

Structure allows recursive decomposition ) enumeration [Tutte, ’60s].

Euler Formula : # vertices+#faces = 2+#edges

A triangulation with 2n faces has 3n edges and n+ 2 vertices.

Some families of maps

Simple triangulations

4-regular maps

Bipolar orientations

Some families of maps

Simple triangulations

4-regular maps

Bipolar orientations

Number of simple triangulations with n+ 2 vertices:

�n =2(4n� 3)!n!(3n� 1)!

[Tutte ’62], [Poulalhon, Schae↵er 05]

Some families of maps

Simple triangulations

4-regular maps

Bipolar orientations

Number of rooted 4-regular maps with n vertices:

Number of simple triangulations with n+ 2 vertices:

�n =2(4n� 3)!n!(3n� 1)!

[Tutte ’62], [Poulalhon, Schae↵er 05]

[Tutte ’62], [Schae↵er ’97]

Rn =2 · 3n

(n+ 1)

2nn

!

Some families of maps

Simple triangulations

4-regular maps

Bipolar orientationsNumber of bipolar orientations with i+ 2 vertices and j + 1 faces:

[Baxter ’01], [Fusy, Poulalhon, Schae↵er’09],

[Bonichon, Bousquet-Melou, Fusy ’10]

⇥ij =2(i+ j)!(i+ j + 1)!(i+ j + 2)!i!(i+ 1)!(i+ 2)!j!(j + 1)!(j + 2)!

.

Number of rooted 4-regular maps with n vertices:

Number of simple triangulations with n+ 2 vertices:

�n =2(4n� 3)!n!(3n� 1)!

[Tutte ’62], [Poulalhon, Schae↵er 05]

[Tutte ’62], [Schae↵er ’97]

Rn =2 · 3n

(n+ 1)

2nn

!

Enumeration

One of the main question when studying some families of maps:

How many maps belong to this family ?

• Recursive decomposition: Tutte ’60s. Baxter ’01

• Matrix integrals: t’Hooft ’74,Brezin, Itzykson, Parisi and Zuber ’78.

• Representation of the symmetric group: Goulden and Jackson ’87.

• Bijective approach with labeled trees: Cori-Vauquelin ’81, Schae↵er’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy,Miermont, ...

• Bijective approach with blossoming trees: Schae↵er ’98, Schae↵erand Bousquet-Melou ’00, Poulalhon and Schae↵er ’05, Fusy, Poulalhonand Schae↵er ’06.

Enumeration

One of the main question when studying some families of maps:

How many maps belong to this family ?

• Recursive decomposition: Tutte ’60s. Baxter ’01

• Matrix integrals: t’Hooft ’74,Brezin, Itzykson, Parisi and Zuber ’78.

• Representation of the symmetric group: Goulden and Jackson ’87.

• Bijective approach with labeled trees: Cori-Vauquelin ’81, Schae↵er’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy,Miermont, ...

• Bijective approach with blossoming trees: Schae↵er ’98, Schae↵erand Bousquet-Melou ’00, Poulalhon and Schae↵er ’05, Fusy, Poulalhonand Schae↵er ’06.

Bijective proofs ?

• Bijective approach with labeled trees: Cori-Vauquelin ’81, Schae↵er’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy,Miermont, ...

• Bijective approach with blossoming trees: Schae↵er ’98, Schae↵erand Bousquet-Melou ’00, Poulalhon and Schae↵er ’05, Fusy, Poulalhonand Schae↵er ’06.

Bijective proofs ?

• Bijective approach with labeled trees: Cori-Vauquelin ’81, Schae↵er’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy,Miermont, ...

• Bijective approach with blossoming trees: Schae↵er ’98, Schae↵erand Bousquet-Melou ’00, Poulalhon and Schae↵er ’05, Fusy, Poulalhonand Schae↵er ’06.

Bijective proof give enumeration formulas but they also yield:

• an explanation of why the formulas are so simple.

• fast algorithms to encode/generate random maps

• better understanding of the structure of the map.

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

A plane map can be canonically associated to any blossoming tree bymaking all closures clockwise.

What is a blossoming tree ?

If the tree is rooted and its edges oriented towards the root + closureedges oriented naturally

) Accessible orientation of the map without ccw cycles.

A plane map can be canonically associated to any blossoming tree bymaking all closures clockwise.

What is a blossoming tree ?

If the tree is rooted and its edges oriented towards the root + closureedges oriented naturally

) Accessible orientation of the map without ccw cycles.

A plane map can be canonically associated to any blossoming tree bymaking all closures clockwise.

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Proof by induction onthe number of faces +identification of closureedges ....

Orientations

To apply the construction: need to find canonical orientations

Orientation = orientation of the edges of the map

Orientations

Bipolar orientations

To apply the construction: need to find canonical orientations

canonical orientation =

acyclic + accessible

Orientation = orientation of the edges of the map

Orientations

Bipolar orientations

To apply the construction: need to find canonical orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Orientation = orientation of the edges of the map

Orientations

Bipolar orientations

To apply the construction: need to find canonical orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Orientation = orientation of the edges of the map

A map is 4-regular i↵ it admits an orientation withindegree 2 and outdegree 2 for each vertex.

Orientations

Bipolar orientations

To apply the construction: need to find canonical orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Orientation = orientation of the edges of the map

Orientations

Bipolar orientations

To apply the construction: need to find canonical orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Orientation = orientation of the edges of the map

A triangulation is simple i↵ it admits an orientation with:outdegree 3 for each non-root vertex,outdegree 1 for each vertex on the root face.

Orientations

Many families of maps admit a caracterization via orientations

Bipolar orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Orientations

Bipolar orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Theorem requires accessible orientation without ccw cycles :Too much to ask ?

Orientations

Bipolar orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Theorem requires accessible orientation without ccw cycles :Too much to ask ?

NO !

Orientations

Bipolar orientations

canonical orientation =

acyclic + accessible

4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex

Simple triangulations

3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex

Theorem requires accessible orientation without ccw cycles :Too much to ask ?

NO !

Proposition: [Felsner ’04]For a given map and orientation, there exists a unique orientation withthe same outdegrees and without ccw cycles.If there exists one accessible ↵-orientation, all of them are accessible.

Summary

• Take a family of maps,

Maps with even degrees

Summary

• Take a family of maps,

Maps with even degrees

• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degrees

Summary

• Take a family of maps,

Maps with even degrees

• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degrees

• Consider the unique orientation without counterclockwise cycles,

Summary

• Take a family of maps,

Maps with even degrees

• Consider the unique orientation without counterclockwise cycles,• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degrees

Summary

• Take a family of maps,

Maps with even degrees

• Consider the unique orientation without counterclockwise cycles,• Apply the bijection,

• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degrees

Summary

• Take a family of maps,

Maps with even degrees

• Consider the unique orientation without counterclockwise cycles,• Apply the bijection,• Study the family of blossoming trees.

• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degreesTrees with same out/indegrees

Blossoming trees and bipolar orientations

s

t

Blossoming trees and bipolar orientations

s

t

s

t

Blossoming trees and bipolar orientations

s

t

s

t

Blossoming trees and bipolar orientations

s

t

s

t

Blossoming trees and bipolar orientations

s

t

s

t

Blossoming trees and bipolar orientations

s

t t

marked vertex 2 outer face ) easy to compute the blossoming tree

[Bernardi ’07]

Blossoming trees and bipolar orientations

s

t t

marked vertex 2 outer face ) easy to compute the blossoming tree

[Bernardi ’07]Description/enumeration of these trees ?

Bijection

Blossoming trees and triplet of paths

blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces

Tbip(i, j) =

Blossoming trees and triplet of paths

blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces

Tbip(i, j) =

Theorem

=blossoming trees which closes into a bipolar orientationwith i+ 2 vertices and j + 1 faces

Blossoming trees and triplet of paths

blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces

Tbip(i, j) =

Theorem

Proposition: [A., Poulalhon]There exists a one-to-one correspondence between :

Tbip(i, j) and

triplet of non-intersecting paths

with i and j

and fixed first and final points

=blossoming trees which closes into a bipolar orientationwith i+ 2 vertices and j + 1 faces

Blossoming trees and triplet of paths

blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces

Tbip(i, j) =

Proposition: [A., Poulalhon]There exists a one-to-one correspondence between :

Tbip(i, j) and

triplet of non-intersecting paths

with i and j

and fixed first and final points

Lindstrom-Gessel-Viennot Lemma

#Tbip(i, j) = ⇥ij =2(i+j)!(i+j+1)!(i+j+2)!

i!(i+1)!(i+2)!j!(j+1)!(j+2)!

Trees of Tbip

Trees of Tbip

• First son of the root = only opening stems

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

• Leaf of the tree ( 6= s) must carry one closing stem.

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

• Leaf of the tree ( 6= s) must carry one closing stem.

Encoding of the blossoming tree = contour word = word on {e, e, b, b} s.t.:

e, e : first time, second time we see an edgeb, b : opening stem, closing stem.

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

• Leaf of the tree ( 6= s) must carry one closing stem.

Encoding of the blossoming tree = contour word = word on {e, e, b, b} s.t.:

e, e : first time, second time we see an edgeb, b : opening stem, closing stem.

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

) w = eb

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

• Leaf of the tree ( 6= s) must carry one closing stem.

Encoding of the blossoming tree = contour word = word on {e, e, b, b} s.t.:

e, e : first time, second time we see an edgeb, b : opening stem, closing stem.

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

) w = eb

) w[b, b] = Dyck word

Trees of Tbip

• First son of the root = only opening stems

• Closures must not wrap the root

• Opening stem cannot close into its subtree

• Leaf of the tree ( 6= s) must carry one closing stem.

Encoding of the blossoming tree = contour word = word on {e, e, b, b} s.t.:

e, e : first time, second time we see an edgeb, b : opening stem, closing stem.

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

) w = eb

) w[b, b] = Dyck word

) w = e . . . e . . . b . . . e . . .

) w = . . . b . . . e . . . b . . . 9

9

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

w1 := w[e, b], w2 := w[e, b]

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

w1 := w[e, b], w2 := w[e, b]

w1 = e e e e b e b b e b b e b bw2 = e b e b e e b e b e b b b e

e

e

b

b

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

w1 := w[e, b], w2 := w[e, b]

w1 = e e e e b e b b e b b e b bw2 = e b e b e e b e b e b b b e

e

e

b

b • w2 = on the right/bottom of w1

• no vertical edge in common.

• w1 = e . . . b w2 = eb

+

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

w1 := w[e, b], w2 := w[e, b]

w1 = e e e e b e b b e b b e b bw2 = e b e b e e b e b e b b b e

e

e

b

b • w2 = on the right/bottom of w1

• no vertical edge in common.

• w1 = e . . . b w2 = eb

+

e

eb

b

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .

w1 := w[e, b], w2 := w[e, b]

w1 = e e e e b e b b e b b e b bw2 = e b e b e e b e b e b b b e

e

e

b

b • w2 = on the right/bottom of w1

• no vertical edge in common.

• w1 = e . . . b w2 = eb

+

e

eb

b

+ w3 = w[e, b] = triple of paths !

Summary

Proposition: [A., Poulalhon]There exists a one-to-one correspondence between :

Tbip(i, j) and

triplet of non-intersecting paths

with i and j

and fixed first and final points

Corollary : The number ⇥ij of bipolar orientationswith i+ 2 vertices and j + 1 faces is equal to:

⇥ij =2(i+ j)!(i+ j + 1)!(i+ j + 2)!

i!(i+ 1)!(i+ 2)!j!(j + 1)!(j + 2)!.

Conclusion + perspectives

Our framework can be applied to many families of maps :• Simple triangulations and quadrangulations• Eulerian and general maps• Non-separable maps• Constellations• Plane bipolar orientations• d-angulations of girth d• . . .

New bijectionsNew bijections

simpler proofs ofknown bijections

Conclusion + perspectives

Our framework can be applied to many families of maps :• Simple triangulations and quadrangulations• Eulerian and general maps• Non-separable maps• Constellations• Plane bipolar orientations• d-angulations of girth d• . . .

New bijectionsNew bijections

simpler proofs ofknown bijections

Unified point of view + can work even if no closed formulas

Conclusion + perspectives

Our framework can be applied to many families of maps :• Simple triangulations and quadrangulations• Eulerian and general maps• Non-separable maps• Constellations• Plane bipolar orientations• d-angulations of girth d• . . .

New bijectionsNew bijections

simpler proofs ofknown bijections

Unified point of view + can work even if no closed formulas

Di↵erent version of a unified bijection in [Bernardi, Fusy ’10],

• Can we make a connection between both ?

• Can we understand the bijections better: distance, ...

Conclusion + perspectives

Our framework can be applied to many families of maps :• Simple triangulations and quadrangulations• Eulerian and general maps• Non-separable maps• Constellations• Plane bipolar orientations• d-angulations of girth d• . . .

New bijectionsNew bijections

simpler proofs ofknown bijections

Thank you !

Unified point of view + can work even if no closed formulas

Di↵erent version of a unified bijection in [Bernardi, Fusy ’10],

• Can we make a connection between both ?

• Can we understand the bijections better: distance, ...