Bijective counting of tree-rooted maps

Transcription

1 Bijective counting of tree-rooted maps Olivier Bernardi - LaBRI, Bordeaux Combinatorics and Optimization seminar, March 2006, Waterloo University

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. Olivier Bernardi - LaBRI p.1/31

3 Maps and trees Olivier Bernardi - LaBRI p.2/31

4 Planar maps A map is a connected planar graph properly embedded in the oriented sphere. The map is considered up to deformation. = Olivier Bernardi - LaBRI p.3/31

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. Olivier Bernardi - LaBRI p.3/31

6 Trees A tree is a map with only one face. Olivier Bernardi - LaBRI p.4/31

7 Trees A tree is a map with only one face. The size of a map, a tree, is the number of edges. Olivier Bernardi - LaBRI p.4/31

8 Tree-rooted maps A submap is a spanning tree if it is a tree containing every vertex. Olivier Bernardi - LaBRI p.5/31

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. Olivier Bernardi - LaBRI p.5/31

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

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. Olivier Bernardi - LaBRI p.7/31

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. Olivier Bernardi - LaBRI p.8/31

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. Olivier Bernardi - LaBRI p.8/31

14 Trees and parenthesis systems Rooted trees of size n are in bijection with parenthesis systems of size n. Olivier Bernardi - LaBRI p.9/31

15 Trees and parenthesis systems We turn around the tree and write : a the first time we follow an edge, a the second time. aaa Olivier Bernardi - LaBRI p.9/31

16 Trees and parenthesis systems We turn around the tree and write : a the first time we follow an edge, a the second time. aaaaaaaa Olivier Bernardi - LaBRI p.9/31

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. Olivier Bernardi - LaBRI p.10/31

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. Olivier Bernardi - LaBRI p.10/31

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. Olivier Bernardi - LaBRI p.10/31

20 Counting results There are C k = 1 ( ) 2k k + 1 k k. parenthesis systems of size Olivier Bernardi - LaBRI p.11/31

21 Counting results There are C k = 1 ( ) 2k parenthesis systems of size k + 1 k k. ( ) 2n There are ways of shuffling a parenthesis system 2k of size k (on {a, a}) and a parenthesis system of size n k (on {b, b}). Olivier Bernardi - LaBRI p.11/31

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

23 Counting results M n = = = n ( ) 2n C k C n k 2k k=0 (2n)! n ( )( ) n + 1 n + 1 (n + 1)! 2 k n k ( k=0 ) (2n)! 2n + 2 (n + 1)! 2 n Olivier Bernardi - LaBRI p.12/31

24 Counting results M n = = = n ( ) 2n C k C n k 2k k=0 (2n)! n ( )( ) n + 1 n + 1 (n + 1)! 2 k n k ( k=0 ) (2n)! 2n + 2 (n + 1)! 2 n Theorem : The number of parenthesis shuffles of size n is M n = C n C n+1. Olivier Bernardi - LaBRI p.12/31

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

26 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? Olivier Bernardi - LaBRI p.13/31

27 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. Olivier Bernardi - LaBRI p.13/31

28 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? Olivier Bernardi - LaBRI p.13/31

29 Tree-rooted maps Trees Non-crossing partitions Olivier Bernardi - LaBRI p.14/31

30 Orientations of tree-rooted maps Olivier Bernardi - LaBRI p.15/31

31 Orientations of tree-rooted maps Internal edges are oriented from the root to the leaves. Olivier Bernardi - LaBRI p.15/31

32 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. Olivier Bernardi - LaBRI p.15/31

33 Orientations of tree-rooted maps Proposition : The orientation is root-connected : there is an oriented path from the root to any vertex. Olivier Bernardi - LaBRI p.16/31

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. Olivier Bernardi - LaBRI p.16/31

35 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. Olivier Bernardi - LaBRI p.16/31

36 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. Olivier Bernardi - LaBRI p.17/31

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. Olivier Bernardi - LaBRI p.17/31

38 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

39 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

40 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

41 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

42 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

43 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

44 From the orientation to the tree We turn around the tree we are constructing. Olivier Bernardi - LaBRI p.18/31

45 Vertex explosion Olivier Bernardi - LaBRI p.19/31

46 Vertex explosion We explode the vertex and obtain a vertex per ingoing edge + a (gluing) cell. Olivier Bernardi - LaBRI p.19/31

47 Example Olivier Bernardi - LaBRI p.20/31

48 Example Olivier Bernardi - LaBRI p.20/31

49 Example A tree! Olivier Bernardi - LaBRI p.20/31

50 Bijection Proposition : The map obtained by exploding the vertices is a tree. Olivier Bernardi - LaBRI p.21/31

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. Olivier Bernardi - LaBRI p.21/31

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. Olivier Bernardi - LaBRI p.22/31

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 : M n = C n C n+1. Olivier Bernardi - LaBRI p.22/31

54 Example Olivier Bernardi - LaBRI p.23/31

55 Example Olivier Bernardi - LaBRI p.23/31

56 Example Olivier Bernardi - LaBRI p.23/31

57 Example Olivier Bernardi - LaBRI p.23/31

58 Example Olivier Bernardi - LaBRI p.23/31

59 Isomorphism with a bijection by Cori, Dulucq and Viennot Olivier Bernardi - LaBRI p.24/31

60 Tree code Φ Definition : Φ(ɛ) = u v. Φ a : Replace last occurrence of u by u v. Φ b : Replace first occurrence of v by u v. Φ a : Replace first occurrence of v by T 1 a v T 2 T 2 T 1 Φ b : Replace last occurrence of u by T 1 b u T2 T 2 T 1 Olivier Bernardi - LaBRI p.25/31

61 Tree code Φ Example : baaaba u v b u u v a u u v v a u u v a u u v v b u v v a u v Olivier Bernardi - LaBRI p.26/31

62 Tree code Φ Example : baaaba u v b u u v a u u v v a u u v a u u v v b u v v a u v baaaba Φ Olivier Bernardi - LaBRI p.26/31

63 Partition code Ψ Definition : Ψ(ɛ) : Ψ a : Replace last active left leaf a Ψ b : Replace first active right leaf b Ψ a : Inactivate first active right leaf. Ψ b : Inactivate last active left leaf. Olivier Bernardi - LaBRI p.27/31

64 Partition code Ψ Example : baaaba b a a a b a Olivier Bernardi - LaBRI p.28/31

65 Partition code Ψ Example : baaaba b a a a b a baaaba Ψ Olivier Bernardi - LaBRI p.28/31

66 Isomorphism Id Θ baaaba Olivier Bernardi - LaBRI p.29/31

67 Isomorphism tree code : Olivier Bernardi - LaBRI p.30/31

68 Isomorphism tree code : Olivier Bernardi - LaBRI p.30/31

69 Isomorphism tree code : u v Olivier Bernardi - LaBRI p.30/31

70 Isomorphism tree code : u u v Olivier Bernardi - LaBRI p.30/31

71 Isomorphism tree code : u u v v Olivier Bernardi - LaBRI p.30/31

72 Isomorphism tree code : u u v v v Olivier Bernardi - LaBRI p.30/31

73 Isomorphism tree code : u v v v Olivier Bernardi - LaBRI p.30/31

74 Isomorphism tree code : u v v Olivier Bernardi - LaBRI p.30/31

75 Isomorphism tree code : u u v v Olivier Bernardi - LaBRI p.30/31

76 Isomorphism tree code : u u v Olivier Bernardi - LaBRI p.30/31

77 Isomorphism tree code : u v Olivier Bernardi - LaBRI p.30/31

78 Thanks. Olivier Bernardi - LaBRI p.31/31