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Kreweras Walks and Loopless Triangulations Olivier Bernardi - LaBRI, Bordeaux MIT Combinatorics Seminar, March 2006, Boston Kreweras walks c a b Walks made of West , South and North East steps, starting and ending at the origin and

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SLIDE 1

Kreweras Walks and Loopless Triangulations

Olivier Bernardi - LaBRI, Bordeaux

MIT Combinatorics Seminar, March 2006, Boston

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SLIDE 2

Kreweras walks

a c b

Walks made of West, South and North − East steps, starting and ending at the origin and confined in the first quadrant.

Boston, March 2006 Olivier Bernardi - LaBRI – p.1/32

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SLIDE 3

Kreweras walks

Theorem (Kreweras 65): The number of Kreweras walks

  • f size n (3n steps) is

kn = 4n (n + 1)(2n + 1) 3n n

  • .

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.2/32

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SLIDE 4

Kreweras walks

Theorem (Kreweras 65): The number of Kreweras walks

  • f size n (3n steps) is

kn = 4n (n + 1)(2n + 1) 3n n

  • .

Theorem (Gessel 86): The generating function K(z) =

  • n≥0

knzn is algebraic.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.2/32

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SLIDE 5

Kreweras walks

Theorem (Kreweras 65): The number of Kreweras walks

  • f size n (3n steps) is

kn = 4n (n + 1)(2n + 1) 3n n

  • .

Theorem (Gessel 86): The generating function K(z) =

  • n≥0

knzn is algebraic. [Niederhausen 82, 83, Bousquet-Mélou 05]

Boston, March 2006 Olivier Bernardi - LaBRI – p.2/32

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SLIDE 6

Kreweras walks and cubic maps

Cubic maps and depth-trees. Bijection: Kreweras walk ⇐ ⇒ Cubic map + Depth-tree. Counting Kreweras walks and cubic maps. Open problems.

Boston, March 2006 Olivier Bernardi - LaBRI – p.3/32

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SLIDE 7

Cubic maps and depth-trees

Boston, March 2006 Olivier Bernardi - LaBRI – p.4/32

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SLIDE 8

Maps

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

=

=

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.5/32

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SLIDE 9

Maps

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

=

=

A map is rooted if a half-edge is distinguished as the root.

Boston, March 2006 Olivier Bernardi - LaBRI – p.5/32

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SLIDE 10

Cubic maps

A map is cubic if every vertex has degree 3.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.6/32

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SLIDE 11

Cubic maps

A map is cubic if every vertex has degree 3. We focus on cubic maps without isthmus.

Boston, March 2006 Olivier Bernardi - LaBRI – p.6/32

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SLIDE 12

Cubic maps and triangulations

Cubic maps without isthmus are the dual of loopless triangulations.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.7/32

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SLIDE 13

Cubic maps and triangulations

Cubic maps without isthmus are the dual of loopless triangulations.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.7/32

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SLIDE 14

Cubic maps and triangulations

Cubic maps without isthmus are the dual of loopless triangulations.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.7/32

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SLIDE 15

Cubic maps and triangulations

Cubic maps without isthmus are the dual of loopless triangulations.

Boston, March 2006 Olivier Bernardi - LaBRI – p.7/32

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SLIDE 16

Cubic maps - counting result

Remark: The number of edges of a cubic map is always a multiple of 3. A cubic map of size n has 3n edges, 2n vertices and n + 2 faces.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.8/32

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SLIDE 17

Cubic maps - counting result

Remark: The number of edges of a cubic map is always a multiple of 3. A cubic map of size n has 3n edges, 2n vertices and n + 2 faces. Theorem (Mullin 65, Poulalhon & Schaeffer 03): The number of cubic maps without isthmus of size n is cn = 2n (n + 1)(2n + 1) 3n n

  • = kn

2n .

Boston, March 2006 Olivier Bernardi - LaBRI – p.8/32

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SLIDE 18

Depth-trees

We consider spanning trees of rooted maps.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.9/32

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SLIDE 19

Depth-trees

A spanning tree of a rooted map is a depth-tree if every external edge links a vertex to one of its ancestors.

YES NO

Boston, March 2006 Olivier Bernardi - LaBRI – p.9/32

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SLIDE 20

Counting depth-trees

Theorem: For any cubic map of size n (3n edges), there are 3 · 2n−1 depth-trees.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.10/32

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SLIDE 21

Counting depth-trees

Theorem: For any cubic map of size n (3n edges), there are 3 · 2n−1 depth-trees. Example: n=3

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.10/32

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SLIDE 22

Counting depth-trees

Theorem: For any cubic map of size n (3n edges), there are 2n depth-trees not containing the root. Example: n=3

Boston, March 2006 Olivier Bernardi - LaBRI – p.10/32

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SLIDE 23

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS).

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 24

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrllr

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 25

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 26

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

r

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 27

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rr

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 28

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrl

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 29

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrll

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 30

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrll

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 31

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrll

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 32

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrll

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 33

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrllr

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.11/32

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SLIDE 34

Counting depth-trees

(Idea of the) proof: The depth-trees are the trees that can be obtained by a depth-first search algorithm (DFS). During a DFS, there are n real binary choices. (One for each external edge.)

rrllr

Boston, March 2006 Olivier Bernardi - LaBRI – p.11/32

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SLIDE 35

Kreweras walk ⇐ ⇒ Cubic map + Depth-tree

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.12/32

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SLIDE 36

Kreweras walk ⇐ ⇒ Cubic map + Depth-tree kn = cn × 2n

Boston, March 2006 Olivier Bernardi - LaBRI – p.12/32

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SLIDE 37

Preliminary remarks

Kreweras walks are words w on {a, b, c} such that |w|a = |w|b = |w|c, for any prefix w′, |w′|a ≤ |w′|c and |w′|b ≤ |w′|c. c w = caccbbcbcbbaaaa b a

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.13/32

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SLIDE 38

Preliminary remarks

Kreweras walks are words w on {a, b, c} such that |w|a = |w|b = |w|c, for any suffix w′, |w′|a ≥ |w′|c and |w′|b ≥ |w′|c. c w = caccbbcbcbbaaaa b a

Boston, March 2006 Olivier Bernardi - LaBRI – p.13/32

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SLIDE 39

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 40

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 41

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 42

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 43

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 44

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 45

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 46

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 47

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 48

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 49

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 50

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 51

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 52

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.14/32

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SLIDE 53

Bijection

Example: w = caccbbcbcbbaaaa

Boston, March 2006 Olivier Bernardi - LaBRI – p.14/32

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SLIDE 54

Theorem: This construction is a bijection between Kreweras walks of size n and cubic maps of size n + depth-tree. Corollary: kn = cn × 2n.

rrrrl

Boston, March 2006 Olivier Bernardi - LaBRI – p.15/32

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SLIDE 55

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 56

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 57

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 58

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 59

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 60

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 61

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 62

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 63

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 64

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 65

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 66

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.16/32

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SLIDE 67

Proof: The reverse bijection

w = caccbbcbcbbaaaa

Boston, March 2006 Olivier Bernardi - LaBRI – p.16/32

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SLIDE 68

Counting Kreweras walks and cubic maps

Boston, March 2006 Olivier Bernardi - LaBRI – p.17/32

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SLIDE 69

Relaxing some constraints

Kreweras walks are the words w on {a, b, c} such that |w|a = |w|b = |w|c, for any prefix w′, |w′|a ≤ |w′|c and |w′|b ≤ |w′|c.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.18/32

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SLIDE 70

Relaxing some constraints

Kreweras walks are the words w on {a, b, c} such that |w|a = |w|b = |w|c, for any prefix w′, |w′|a ≤ |w′|c and |w′|b ≤ |w′|c. What about words w on {a, b, c} such that |w|a + |w|b = 2|w|c, for any prefix w′, |w′|a + |w′|b ≤ 2|w′|c ? We call them extended Kreweras walks.

Boston, March 2006 Olivier Bernardi - LaBRI – p.18/32

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SLIDE 71

Kreweras Extended Kreweras w = caccaacbcbbaaaa w = caccbbcbcbbaaaa |w′|a ≤ |w′|c and |w′|b ≤ |w′|c |w′|a + |w′|b ≤ 2|w′|c

Boston, March 2006 Olivier Bernardi - LaBRI – p.19/32

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SLIDE 72

Proposition: There are en = 4n 2n + 1 3n n

  • extended walks
  • f size n.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.20/32

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SLIDE 73

Proposition: There are en = 4n 2n + 1 3n n

  • extended walks
  • f size n.

1 2n + 1 3n n

  • 4n

Boston, March 2006 Olivier Bernardi - LaBRI – p.20/32

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SLIDE 74

Proof: The extended walks w are such that: |w|a + |w|b = 2|w|c, for all prefix w′, |w′|a + |w′|b ≤ 2|w′|c. Position of the c’s: 1 2n + 1 3n n

  • .

Cycle lemma: There are 1 2n + 1 3n n

  • (one-dimensional)

walks with 3n steps +2 and -1. Position of the a’s and b’s: 22n.

Boston, March 2006 Olivier Bernardi - LaBRI – p.21/32

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SLIDE 75

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

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SLIDE 76

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-77
SLIDE 77

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

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SLIDE 78

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-79
SLIDE 79

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-80
SLIDE 80

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-81
SLIDE 81

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-82
SLIDE 82

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-83
SLIDE 83

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-84
SLIDE 84

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-85
SLIDE 85

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.22/32

slide-86
SLIDE 86

Extending the bijection

Example: w = caccaacbcbbaaaa

Boston, March 2006 Olivier Bernardi - LaBRI – p.22/32

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SLIDE 87

Theorem: This construction is a bijection between extended Kreweras walks of size n and cubic maps of size n + depth-tree + marked external edge. Corollary: en = cn × 2n × (n + 1).

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.23/32

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SLIDE 88

Theorem: This construction is a bijection between extended Kreweras walks of size n and cubic maps of size n + depth-tree + marked external edge. Corollary: en = cn × 2n × (n + 1). Thus, cn = 2n (n + 1)(2n + 1) 3n n

  • and kn =

4n (n + 1)(2n + 1) 3n n

  • .

Boston, March 2006 Olivier Bernardi - LaBRI – p.23/32

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SLIDE 89

Concluding remarks

Boston, March 2006 Olivier Bernardi - LaBRI – p.24/32

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SLIDE 90

Results

We counted depth-trees on cubic maps.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.25/32

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SLIDE 91

Results

We counted depth-trees on cubic maps. We established a bijection between Kreweras walks and cubic maps with a depth-tree. ⇒ Coding of triangulations with log2(27) bits per vertex. (Optimal coding: log2(27) − 1 bits per vertex.)

Boston, March 2006 Olivier Bernardi - LaBRI – p.25/32

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SLIDE 92

Results

We extended the bijection to a more general class of walks. ⇒ Counting results. ⇒ Random sampling of triangulations in linear time. kn = 4n (n + 1)(2n + 1) 3n n

  • .

Boston, March 2006 Olivier Bernardi - LaBRI – p.26/32

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SLIDE 93

Open problems

Is it possible to describe the conjugacy class of a Kreweras walk without using the cubic map ? kn = 4n (n + 1)(2n + 1) 3n n

  • .

Boston, March 2006 Olivier Bernardi - LaBRI – p.27/32

slide-94
SLIDE 94

Open problems

Can we count Kreweras walks ending at (i, 0) ? at (i, j) ?

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.28/32

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SLIDE 95

Open problems

Can we count Kreweras walks ending at (i, 0) ? at (i, j) ? Theorem [Kreweras 65] : kn,i = 4n 2i i

  • 2i + 1

(n + i + 1)(2n + 2i + 1) 3n + 2i n

  • .

Boston, March 2006 Olivier Bernardi - LaBRI – p.28/32

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SLIDE 96

Open problems

Kreweras walks ending at (i, 0) are in bijection with (i + 2)-near-cubic maps + depth-trees. Corollary: kn,i = 2n × cn,i.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.29/32

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SLIDE 97

Open problems

Kreweras walks ending at (i, 0) are in bijection with (i + 2)-near-cubic maps + depth-trees. We just have to begin with i free legs.

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.29/32

slide-98
SLIDE 98

Open problems

Kreweras walks ending at (i, 0) are in bijection with (i + 2)-near-cubic maps + depth-trees. We just have to begin with i free legs. But can we count extended walks ?

Boston, March 2006 Olivier Bernardi - LaBRI – p.29/32

slide-99
SLIDE 99

Open problems

Can we extend this bijection to some other class of maps ? To quadrangulations ?

Boston, March 2006 Olivier Bernardi - LaBRI – p.30/32

slide-100
SLIDE 100

Open problems

There are similar counting results:

  • Non-separable maps [Tutte].
  • Two-stack sortable permutations [West, Zeilberger].

NSn = 2 (n + 1)(2n + 1) 3n n

  • .

Boston, March 2006

▽Olivier Bernardi - LaBRI – p.31/32

slide-101
SLIDE 101

Open problems

There are similar counting results:

  • Non-separable maps [Tutte].
  • Two-stack sortable permutations [West, Zeilberger].

NSn = 2 (n + 1)(2n + 1) 3n n

  • .

[Dulucq, Gire & Guibert 96, Goulden & West 96]

Boston, March 2006 Olivier Bernardi - LaBRI – p.31/32

slide-102
SLIDE 102

Thanks.

Boston, March 2006 Olivier Bernardi - LaBRI – p.32/32