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Unified construction betweenblossoming trees and planar maps
Shanghai Jiao Tong University, November 23rd 2013
Marie Albenque (CNRS, Paris)Joint work with Dominique Poulalhon
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A plane map is the embedding of a connected graph in the planeup to continuous deformations.
Plane Maps.
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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.
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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.
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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.
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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.
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Some families of maps
Simple triangulationsfaces are trianglesno loops nor multiple edges
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Some families of maps
Simple triangulationsfaces are trianglesno loops nor multiple edges
4-regular mapsall vertices have degree 4
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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
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Why maps ?
What the motivation for studying maps instead of graphs ?
Because maps have more structure than graphs,they are actually simpler to study.
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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.
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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.
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Some families of maps
Simple triangulations
4-regular maps
Bipolar orientations
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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]
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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
!
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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
!
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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.
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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 ?
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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 ?
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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 ?
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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 ....
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Orientations
To apply the construction: need to find canonical orientations
Orientation = orientation of the edges of the map
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Orientations
Bipolar orientations
To apply the construction: need to find canonical orientations
canonical orientation =
acyclic + accessible
Orientation = orientation of the edges of the map
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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
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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.
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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
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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.
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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
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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 ?
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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 !
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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.
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Summary
• Take a family of maps,
Maps with even degrees
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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
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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,
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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
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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
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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
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Blossoming trees and bipolar orientations
s
t
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Blossoming trees and bipolar orientations
s
t
s
t
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Blossoming trees and bipolar orientations
s
t
s
t
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Blossoming trees and bipolar orientations
s
t
s
t
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Blossoming trees and bipolar orientations
s
t
s
t
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Blossoming trees and bipolar orientations
s
t t
marked vertex 2 outer face ) easy to compute the blossoming tree
[Bernardi ’07]
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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
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Blossoming trees and triplet of paths
blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces
Tbip(i, j) =
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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
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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
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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)!
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Trees of Tbip
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Trees of Tbip
• First son of the root = only opening stems
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Trees of Tbip
• First son of the root = only opening stems
• Closures must not wrap the root
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Trees of Tbip
• First son of the root = only opening stems
• Closures must not wrap the root
• Opening stem cannot close into its subtree
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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.
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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
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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
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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
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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
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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
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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]
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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
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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
+
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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
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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 !
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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)!.
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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
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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
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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, ...
![Page 82: Unified construction between blossoming trees and …math.sjtu.edu.cn/conference/Bannai/2013/data/20131123A/...Unified construction between blossoming trees and planar maps Shanghai](https://reader034.vdocument.in/reader034/viewer/2022050220/5f65c6534652523e414fd9d2/html5/thumbnails/82.jpg)
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, ...