Binary trees, super-catalan numbers and 3-connected Planar Graphs

Transcription

1 Bnary trees, super-catalan numbers and 3-connected Planar Graphs Glles Schaeffer LIX, CNRS/École Polytechnque Based n part on a jont work wth E. Fusy and D. Poulalhon

2 Mon premer souvenr d un cours de combnatore... Mots de Dyck n+( 2n n ) Arbres bnares Arbres planares Mots de Lukasevcz Mn-jardn de Catalan (D après photocopes de transparents de Vennot, 993)

3 Today s subject: Super Catalan numbers (Catalan, Gessel) 2 (2n)!(2m)! (n + m)!n!m! These numbers are ntegers for all postve m, n. They deserve a combnatoral nterpretaton! For m =, Catalan numbers: For m = 2, the numbers are: (2n)! n!(n+)! =, 2, 5, 4, 42, 32, 429, , 3, 6, 4, 36, 99, 286, 858,... 6(2n)! (n+2)!n! = 6 n+2 n+( 2n n ). n+( 2n n ). We shall dscuss some nterpretatons for m = 2.

4 More precsely, we am at the followng dagram: 6 n+2 n+( 2n n )

5 A one-page prelmnary...

6 In the Catalan garden, I pluck the... Bnary trees wth n nodes and Dyck paths wth length 2n. They are counted by Catalan numbers: n+( 2n n ) Here s a bjecton: ( ) ( ) ( ) ( ( ) ) Turn around the tree, wrte up or down when enterng or extng a left subtree. ( )

7 Frst nterpretatons: unrooted bnary trees

8 Colors make pctures more fun... Edge-3-colored bnary tree = a bnary tree wth colors on the edge and nodes such that there are two type of nodes: Take a bnary tree Choose colors for the root edge and the root vertex: #{colored tree wth n nodes} = 6 n+( 2n n ).

9 Agrculture hors sol unrooted 3-colored tree = lke a 3-colored bnary tree, but wthout the root... These trees have no symmetres: ndeed symmetres of planar trees must leave the center nvarant. Here the center can be: or: each tree has n + 2 dstnct rootngs. #{unrooted 3-c trees wth n nodes} ) = 6 n+2 n+( 2n n = 6 n+( 2n n )

10 Here s our frst super-cat-structure: 6 n+2 n+( 2n n )

11 An elegant restatment: Trees on the hexagonal lattce (Pppenger & Schlech 03) Up to translaton and rotatons, there s a unque way to embed an unrooted colored tree on the colored hexagonal lattce (possbly wth overlaps). = 6 n+2 n+( 2n n )

12 An elegant restatment: Trees on the hexagonal lattce (Pppenger & Schlech 03) Up to translaton and rotatons, there s a unque way to embed an unrooted colored tree on the colored hexagonal lattce (possbly wth overlaps). = 6 n+2 n+( 2n n )

13 An elegant restatment: Trees on the hexagonal lattce (Pppenger & Schlech 03) Up to translaton and rotatons, there s a unque way to embed an unrooted colored tree on the colored hexagonal lattce (possbly wth overlaps). = 6 n+2 n+( 2n n ) = #{hexagonal trees wth n nodes}.

14 We got an arrow! 6 n+2 n+( 2n n )

15 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l

16 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: 0

17 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: 0

18 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: 0-2

19 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: 0-2 -

20 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple:

21 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: More generally:

22 Hexagonal trees are Mrelle s embedded trees... Turn counterclockwse around the tree, and label each sde of edges: at each corner l=l+ at each leaf: l=l Exemple: 0-3 Read Mrelle s labels on the left of nner edges These labels should be vewed as angles (multples of π/3). After a full turn around the tree, the angle varaton s 2π = 6 (π/3). 0-3 More generally:

23 A bgger example

24 A bgger example. 0-3 Rerootng changes the actual label, but not the varatons!

25 A bgger example. 0 3 Rerootng changes the actual label, but not the varatons!

26 A bgger example. 0 3 Rerootng changes the actual label, but not the varatons! By rerootng, the sequence of varatons are cyclcally permuted apply the cycle lemma. 6 n+2 n+( 2n n )

27 Ths should gve Mrelle s formula for postve bnary trees: recall Theorem (part of her)let B n be the number of rooted bnary trees wth n nodes wth label 0. Then B 0 n + B 0 n+ = 6(2n)! n!(n + 2)!. In other terms: B 0 n + B n = 6(2n)! n!(n + 2)!.

28 We got an arrow! 6 n+2 n+( 2n n )

29 Second nterpretaton: Dyck paths

30 Let a Gessel-Xn par be a par of Dyck paths such that the heght of the two paths dffer at most by one. Example: n = n = 2 n = 3 + symmetrc 2 3 6

31 Let a Gessel-Xn par be a par of Dyck paths such that the heght of the two paths dffer at most by one. Example: n = n = 2 n = 3 + symmetrc Theorem (Gessel & Xn). The number of Gessel-Xn pars wth total length 2n s: 4C n C n+ = 6(2n)! (n+2)!n!.

32 Let a Gessel-Xn par be a par of Dyck paths such that the heght of the two paths dffer at most by one. Example: n = n = 2 n = 3 + symmetrc Theorem (Gessel & Xn). The number of Gessel-Xn pars wth total length 2n s: 4C n C n+ = 6(2n)! (n+2)!n!. Can we relate ths to the prevous bnary trees?

33 Decomposton at the center of the tree peelng

34 Decomposton at the center of the tree peelng Two bnary trees wth equal heght: k T k(z) 2

35 Decomposton at the center of the tree peelng Two bnary trees wth equal heght: k T k(z) 2 The center can also be a node:

36 Decomposton at the center of the tree peelng Two bnary trees wth equal heght: k T k(z) 2 The center can also be a node: Two bnary trees wth almost the same heght: k T k(z)t k (z)

37 Decomposton at the center of the tree peelng Two bnary trees wth equal heght: k T k(z) 2 The center can also be a node: Two bnary trees wth almost the same heght: k T k(z)t k (z) But ths approach does not yeld the relaton to Dyck paths: Colors are not taken nto account correctly... Not the rght noton of heght!

38 A noton of center nherted from Dyck paths. 2 ( ) ( ) ( ) ( ( ) ) ( ) Recall the bjecton... n+( 2n n ) hence the rule for computng the heght: k = max( +, j) k j 0 0 #{ j } j 6 = ( ) n+2 n+( 2n n ).

39 Dependng on the poston of the root, each edge can get two labels: there s a heght labellng of an unrooted tree! Exemple: Theorem. Exactly one of the followng two cases occur: there s one edge wth the 2 labels that are equal, or there s one vertex wth the 3 ncdent labels that are equal.

40 Decomposton at the center of the tree The center s an edge: Two bnary trees wth equal heght: 3 k D k(z) 2 The center s a node: The center can also be a node: Three bnary trees wth the same heght: 2 k zd k(z) 3 Ths s correct: k 3D k(z) 2 + 2zD k (z) 3 = 6(2n)! n!(n+2)! zn. But what we want are pars of Dyck paths wth almost the same heght.

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43 + + + =

44 + = + + +?? Lookng at possble label and exchangng some subtrees, complete the mssng terms!

45 Here s our dagram n n+2 n+( n )

46 Thrd nterpretaton: graphs...

47 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule

48 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule

49 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule

50 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule

51 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule

52 A combnatoral operaton: the local closure Start wth a bnary tree and apply greedly the local closure rule Exactly 6 new vertces are needed

53 A combnatoral operaton: the complete closure Add a hexagon around the pcture

54 A combnatoral operaton: the complete closure Add a hexagon around the pcture Form quadrangles... Ths yelds the quadrangulaton of a hexagon.

55 Theorem (Fusy, Poulalhon, S. 05). The closure s a bjecton between unrooted bnary trees wth n nodes, unrooted quadrangulatons of a hexagon wth n nternal vertces. (I wll not prove ths theorem: t s hard...) Corollary. (Mulln & Schellenberg 68) The number of rooted quadrangulatons of a hexagon s ( ) 6 n + 2 2n. n + n

56 The dagram s almost complete, but we stll mss the 3-connected planar graphs of the ttle of the talk. 6 n+2 n+( 2n n )

57 The dagram s almost complete, but we stll mss the 3-connected planar graphs of the ttle of the talk. 6 n+2 n+( 2n n )

58 Quadrangulatons of a hexagon are almost n bjecton wth 3-connected planar graphs. More precsely: Theorem. (Tutte) There s a smple bjecton between 3-connected planar maps wth n edges, quadrangulatons of a square wth n faces. Theorem (Whtney). 3-connected planar graphs have essentally only one embeddng n the plane.

59 Gessel-Xn pars wth length 2n Unrooted colored trees wth n nodes 6 (n+2)(n+)( 2n n ) 3-connected planar graphs wth n edges Quadrangulatons of a hexagon wth n nner vertces

60 Gessel-Xn pars Unrooted colored trees wth and j 3 (2+)(2j+) ( 2+ j )( 2j+ ) 3-connected planar graphs wth faces and j vertces Quadrangulatons of a hexagon wth + 3 and j + 3

61 (order super) Catalan order 2 super Catalan unvarate (2n)! n!(n+)! 6(2n)! n!(n+2)! bvarate (2+)!(2j)!!j!(2+ j)!(2j+ )! 3(2)!(2j)!!j!(2+ j)!(2j+ )! (m, n) super Catalan 2 (2n)!(2m)! n!m!(n+m)!??? Maybe havng a 2-varable verson could help fndng a combnatoral nterpretaton for all (m, n)...

62 (order super) Catalan order 2 super Catalan unvarate (2n)! n!(n+)! 6(2n)! n!(n+2)! bvarate (2+)!(2j)!!j!(2+ j)!(2j+ )! 3(2)!(2j)!!j!(2+ j)!(2j+ )! (m, n) super Catalan 2 (2n)!(2m)! n!m!(n+m)!??? Maybe havng a 2-varable verson could help fndng a combnatoral nterpretaton for all (m, n)... That s all. Merc de votre attenton!

63 Gessel-Xn pars wth length 2n Unrooted colored trees wth n nodes 6 (n+2)(n+)( 2n n ) 3-connected planar graphs wth n edges Quadrangulatons of a hexagon wth n nner vertces

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