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Bijective counting of tree-rooted maps Olivier Bernardi - LaBRI, - - PowerPoint PPT Presentation

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Bijective counting of tree-rooted maps Olivier Bernardi - LaBRI, Bordeaux Combinatorics and Optimization seminar, March 2006, Waterloo University Bijective counting of tree-rooted maps Maps and trees. Tree-rooted maps and parenthesis systems.

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SLIDE 1

Bijective counting of tree-rooted maps

Olivier Bernardi - LaBRI, Bordeaux

Combinatorics and Optimization seminar, March 2006, Waterloo University

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SLIDE 2

Bijective counting of tree-rooted maps

Maps and trees. Tree-rooted maps and parenthesis systems. (Mullin, Lehman & Walsh) Bijection : Tree-rooted maps ⇐ ⇒ Trees × Non-crossing partitions. Isomorphism with a construction by Cori, Dulucq and Viennot.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.1/31

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SLIDE 3

Maps and trees

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.2/31

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SLIDE 4

Planar maps

A map is a connected planar graph properly embedded in the oriented sphere. The map is considered up to deformation.

=

=

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.3/31

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SLIDE 5

Planar maps

A map is a connected planar graph properly embedded in the oriented sphere. The map is considered up to deformation.

=

=

A map is rooted by adding a half-edge in a corner.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.3/31

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SLIDE 6

Trees

A tree is a map with only one face.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.4/31

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SLIDE 7

Trees

A tree is a map with only one face. The size of a map, a tree, is the number of edges.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.4/31

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SLIDE 8

Tree-rooted maps

A submap is a spanning tree if it is a tree containing every vertex.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.5/31

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SLIDE 9

Tree-rooted maps

A submap is a spanning tree if it is a tree containing every vertex. A tree-rooted map is a rooted map with a distinguished spanning tree.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.5/31

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SLIDE 10

Tree-rooted maps and Parenthesis systems (Mullin, Lehman & Walsh)

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.6/31

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SLIDE 11

Parenthesis systems

A parenthesis system is a word w on {a, a} such that |w|a = |w′|a and for all prefix w′, |w′|a ≥ |w′|a. Example : w = aaaaaaaa is a parenthesis system.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.7/31

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SLIDE 12

Parenthesis shuffle

A parenthesis shuffle is a word w on {a, a, b, b} such that the subwords made of {a, a} letters and {b, b} letters are parenthesis systems. Example : w = baababbabaabaa is a parenthesis shuffle.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.8/31

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SLIDE 13

Parenthesis shuffle

A parenthesis shuffle is a word w on {a, a, b, b} such that the subwords made of {a, a} letters and {b, b} letters are parenthesis systems. Example : w = baababbabaabaa is a parenthesis shuffle. The size of a parenthesis system, shuffle is half its length.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.8/31

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SLIDE 14

Trees and parenthesis systems

Rooted trees of size n are in bijection with parenthesis systems of size n.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.9/31

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SLIDE 15

Trees and parenthesis systems

aaa

We turn around the tree and write :

a the first time we follow an edge, a the second time.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.9/31

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SLIDE 16

Trees and parenthesis systems

aaaaaaaa

We turn around the tree and write :

a the first time we follow an edge, a the second time.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.9/31

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SLIDE 17

Tree-rooted maps and parenthesis shuffles

[Mullin 67, Lehman & Walsh 72] Tree-rooted maps of size n are in bijection with parenthesis shuffles of size n.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.10/31

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SLIDE 18

Tree-rooted maps and parenthesis shuffles

baaba

We turn around the tree and write :

a the first time we follow an internal edge, a the second time, b the first time we cross an external edge, b the second time.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.10/31

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SLIDE 19

Tree-rooted maps and parenthesis shuffles

baababbabaabaa

We turn around the tree and write :

a the first time we follow an internal edge, a the second time, b the first time we cross an external edge, b the second time.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.10/31

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SLIDE 20

Counting results

There are Ck = 1 k + 1 2k k

  • parenthesis systems of size

k.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.11/31

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SLIDE 21

Counting results

There are Ck = 1 k + 1 2k k

  • parenthesis systems of size

k. There are 2n 2k

  • ways of shuffling a parenthesis system
  • f size k (on {a, a}) and a parenthesis system of size

n − k (on {b, b}).

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.11/31

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SLIDE 22

Counting results

There are Ck = 1 k + 1 2k k

  • parenthesis systems of size

k. There are 2n 2k

  • ways of shuffling a parenthesis system
  • f size k (on {a, a}) and a parenthesis system of size

n − k (on {b, b}). = ⇒ There are Mn =

n

  • k=0

2n 2k

  • CkCn−k parenthesis

shuffles of size n.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.11/31

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SLIDE 23

Counting results

Mn =

n

  • k=0

2n 2k

  • CkCn−k

= (2n)! (n + 1)!2

n

  • k=0

n + 1 k n + 1 n − k

  • =

(2n)! (n + 1)!2 2n + 2 n

  • Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.12/31

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SLIDE 24

Counting results

Mn =

n

  • k=0

2n 2k

  • CkCn−k

= (2n)! (n + 1)!2

n

  • k=0

n + 1 k n + 1 n − k

  • =

(2n)! (n + 1)!2 2n + 2 n

  • Theorem : The number of parenthesis shuffles of size n is

Mn = CnCn+1.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.12/31

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SLIDE 25

Counting results

Mn =

n

  • k=0

2n 2k

  • CkCn−k

= (2n)! (n + 1)!2

n

  • k=0

n + 1 k n + 1 n − k

  • =

(2n)! (n + 1)!2 2n + 2 n

  • Theorem [Mullin 67] : The number of tree-rooted maps of

size n is Mn = CnCn+1.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.12/31

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A pair of trees ?

Theorem [Mullin 67] : The number of tree-rooted maps of size n is Mn = CnCn+1. Is there a pair of trees hiding somewhere ?

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.13/31

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SLIDE 27

A pair of trees ?

Theorem [Mullin 67] : The number of tree-rooted maps of size n is Mn = CnCn+1. Is there a pair of trees hiding somewhere ? Theorem [Cori, Dulucq, Viennot 86] : There is a (recursive) bijection between parenthesis shuffles of size n and pairs of trees.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.13/31

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SLIDE 28

A pair of trees ?

Theorem [Mullin 67] : The number of tree-rooted maps of size n is Mn = CnCn+1. Is there a pair of trees hiding somewhere ? Theorem [Cori, Dulucq, Viennot 86] : There is a (recursive) bijection between parenthesis shuffles of size n and pairs of trees. Is there a good interpretation on maps ?

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.13/31

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Tree-rooted maps ⇐ ⇒ Trees × Non-crossing partitions

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.14/31

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Orientations of tree-rooted maps

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.15/31

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SLIDE 31

Orientations of tree-rooted maps

Internal edges are oriented from the root to the leaves.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.15/31

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Orientations of tree-rooted maps

Internal edges are oriented from the root to the leaves. External edges are oriented in such a way their heads appear before their tails around the tree.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.15/31

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SLIDE 33

Orientations of tree-rooted maps

Proposition : The orientation is root-connected : there is an oriented path from the root to any vertex.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.16/31

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SLIDE 34

Orientations of tree-rooted maps

Proposition : The orientation is root-connected : there is an oriented path from the root to any vertex. The orientation is minimal : every directed cycle is oriented clockwise.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.16/31

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Orientations of tree-rooted maps

Proposition : The orientation is root-connected : there is an oriented path from the root to any vertex. The orientation is minimal : every directed cycle is oriented clockwise. We call tree-orientation a minimal root-connected

  • rientation.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.16/31

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Orientations of tree-rooted maps

Theorem : The orientation of edges in tree-rooted maps gives a bijection between tree-rooted maps and tree-oriented maps.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.17/31

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SLIDE 37

Orientations of tree-rooted maps

Theorem : The orientation of edges in tree-rooted maps gives a bijection between tree-rooted maps and tree-oriented maps.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.17/31

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SLIDE 38

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 39

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 40

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 41

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 42

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 43

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.18/31

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SLIDE 44

From the orientation to the tree

We turn around the tree we are constructing.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.18/31

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Vertex explosion

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.19/31

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Vertex explosion

We explode the vertex and obtain a vertex per ingoing edge + a (gluing) cell.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.19/31

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Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.20/31

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Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.20/31

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Example

A tree !

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.20/31

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Bijection

Proposition : The map obtained by exploding the vertices is a tree.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.21/31

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SLIDE 51

Bijection

Proposition : The map obtained by exploding the vertices is a tree. The gluing cells are incident to the first corner of each

  • vertex. They define a non-crossing partition of the

vertices of the tree.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.21/31

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SLIDE 52

Bijection

Theorem : The orientation of tree-rooted maps and the explosion of vertices gives a bijection between tree-rooted maps of size n and trees of size n × non-crossing partitions of size n + 1.

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.22/31

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SLIDE 53

Bijection

Theorem : The orientation of tree-rooted maps and the explosion of vertices gives a bijection between tree-rooted maps of size n and trees of size n × non-crossing partitions of size n + 1. Corollary : Mn = CnCn+1.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.22/31

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Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.23/31

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Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.23/31

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Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.23/31

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SLIDE 57

Example

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.23/31

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SLIDE 58

Example

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.23/31

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SLIDE 59

Isomorphism with a bijection by Cori, Dulucq and Viennot

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.24/31

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SLIDE 60

Tree code Φ

Definition : Φ(ǫ) = u • v. Φa : Replace last occurrence of u by u • v. Φb : Replace first occurrence of v by u • v. Φa : Replace first occurrence of v by a v T2 T2 T1 T1 Φb : Replace last occurrence of u by b u T2 T2 T1 T1

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.25/31

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Tree code Φ

Example : baaaba a b b a a a v u v u v u v v u u u u v v u v v u u v u

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.26/31

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SLIDE 62

Tree code Φ

Example : baaaba a b b a a a v u v u v u v v u u u u v v u v v u u v u

Φ baaaba

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.26/31

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SLIDE 63

Partition code Ψ

Definition : Ψ(ǫ) : Ψa : Replace last active left leaf a Ψb : Replace first active right leaf b Ψa : Inactivate first active right leaf. Ψb : Inactivate last active left leaf.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.27/31

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SLIDE 64

Partition code Ψ

Example : baaaba b a a a b a

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.28/31

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SLIDE 65

Partition code Ψ

Example : baaaba b a a a b a

Ψ baaaba

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.28/31

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SLIDE 66

Isomorphism

Id

baaaba Θ

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.29/31

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SLIDE 67

Isomorphism tree code :

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 68

Isomorphism tree code :

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 69

Isomorphism tree code :

u v

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 70

Isomorphism tree code :

u v u

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 71

Isomorphism tree code :

u v u v

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 72

Isomorphism tree code :

v u v u v

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 73

Isomorphism tree code :

u v v v

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 74

Isomorphism tree code :

v v u

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 75

Isomorphism tree code :

v u v u

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 76

Isomorphism tree code :

u v u

Waterloo, March 2006

▽Olivier Bernardi - LaBRI – p.30/31

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SLIDE 77

Isomorphism tree code :

u v

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.30/31

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Thanks.

Waterloo, March 2006 Olivier Bernardi - LaBRI – p.31/31