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

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. (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

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

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

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

Trees A tree is a map with only one face. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.4/31

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

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

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

Tree-rooted maps and Parenthesis systems (Mullin, Lehman & Walsh) Waterloo, March 2006 Olivier Bernardi - LaBRI – p.6/31

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

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

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

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

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

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

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

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

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

Counting results 1 � 2 k � There are C k = parenthesis systems of size k + 1 k k . Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.11/31

Counting results 1 � 2 k � There are C k = parenthesis systems of size k + 1 k k . � 2 n � There are ways of shuffling a parenthesis system 2 k of 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

Counting results 1 � 2 k � There are C k = parenthesis systems of size k + 1 k k . � 2 n � There are ways of shuffling a parenthesis system 2 k of size k (on { a, a } ) and a parenthesis system of size n − k (on { b, b } ). n � 2 n � � = ⇒ There are M n = C k C n − k parenthesis 2 k k =0 shuffles of size n . Waterloo, March 2006 Olivier Bernardi - LaBRI – p.11/31

Counting results n � 2 n � � M n = C k C n − k 2 k k =0 n � n + 1 �� n + 1 � (2 n )! � = ( n + 1)! 2 n − k k k =0 � 2 n + 2 � (2 n )! = ( n + 1)! 2 n Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.12/31

Counting results n � 2 n � � M n = C k C n − k 2 k k =0 n � n + 1 �� n + 1 � (2 n )! � = ( n + 1)! 2 n − k k k =0 � 2 n + 2 � (2 n )! = ( n + 1)! 2 n Theorem : The number of parenthesis shuffles of size n is M n = C n C n +1 . Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.12/31

Counting results n � 2 n � � M n = C k C n − k 2 k k =0 n � n + 1 �� n + 1 � (2 n )! � = ( n + 1)! 2 n − k k k =0 � 2 n + 2 � (2 n )! = ( n + 1)! 2 n Theorem [Mullin 67] : The number of tree-rooted maps of size n is M n = C n C n +1 . Waterloo, March 2006 Olivier Bernardi - LaBRI – p.12/31

A pair of trees ? Theorem [Mullin 67] : The number of tree-rooted maps of size n is M n = C n C n +1 . Is there a pair of trees hiding somewhere ? Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.13/31

A pair of trees ? Theorem [Mullin 67] : The number of tree-rooted maps of size n is M n = C n C n +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

A pair of trees ? Theorem [Mullin 67] : The number of tree-rooted maps of size n is M n = C n C n +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

Tree-rooted maps ⇐ ⇒ Trees × Non-crossing partitions Waterloo, March 2006 Olivier Bernardi - LaBRI – p.14/31

Orientations of tree-rooted maps Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.15/31

Orientations of tree-rooted maps Internal edges are oriented from the root to the leaves. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.15/31

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

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

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

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

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

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

From the orientation to the tree We turn around the tree we are constructing. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.18/31

From the orientation to the tree We turn around the tree we are constructing. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.18/31

From the orientation to the tree We turn around the tree we are constructing. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.18/31

From the orientation to the tree We turn around the tree we are constructing. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.18/31

From the orientation to the tree We turn around the tree we are constructing. Waterloo, March 2006 ▽ Olivier Bernardi - LaBRI – p.18/31