Kreweras Walks and Loopless Triangulations Olivier Bernardi - - PowerPoint PPT Presentation

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.14/32

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.14/32

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.14/32

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.14/32

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.14/32

Bijection Example: w = caccbbcbcbbaaaa Boston, March 2006 Olivier Bernardi - LaBRI – p.14/32

Theorem: This construction is a bijection between Kreweras walks of size n and cubic maps of size n + depth-tree. 2 n . × Corollary: = k n c n rrrrl Boston, March 2006 Olivier Bernardi - LaBRI – p.15/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.16/32

Proof: The reverse bijection w = caccbbcbcbbaaaa Boston, March 2006 Olivier Bernardi - LaBRI – p.16/32

Counting Kreweras walks and cubic maps Boston, March 2006 Olivier Bernardi - LaBRI – p.17/32

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

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

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

4 n � 3 n � Proposition: There are e n = extended walks 2 n + 1 n of size n . Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.20/32

4 n � 3 n � Proposition: There are e n = extended walks 2 n + 1 n of size n . � 3 n � 1 4 n 2 n + 1 n Boston, March 2006 Olivier Bernardi - LaBRI – p.20/32

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 . � 3 n � 1 Position of the c’s : . 2 n + 1 n 1 � 3 n � Cycle lemma: There are (one-dimensional) 2 n + 1 n walks with 3 n steps +2 and -1. Position of the a’s and b’s : 2 2 n . Boston, March 2006 Olivier Bernardi - LaBRI – p.21/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.22/32

Extending the bijection Example: w = caccaacbcbbaaaa Boston, March 2006 Olivier Bernardi - LaBRI – p.22/32

Theorem: This construction is a bijection between extended Kreweras walks of size n and cubic maps of size n + depth-tree + marked external edge. e n = c n × 2 n × ( n + 1) . Corollary: Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.23/32

Theorem: This construction is a bijection between extended Kreweras walks of size n and cubic maps of size n + depth-tree + marked external edge. e n = c n × 2 n × ( n + 1) . Corollary: Thus, 2 n 4 n � 3 n � � 3 n � c n = and k n = . ( n + 1)(2 n + 1) ( n + 1)(2 n + 1) n n Boston, March 2006 Olivier Bernardi - LaBRI – p.23/32

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

Results We counted depth-trees on cubic maps. Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.25/32

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 log 2 (27) bits per vertex. (Optimal coding: log 2 (27) − 1 bits per vertex.) Boston, March 2006 Olivier Bernardi - LaBRI – p.25/32

Results We extended the bijection to a more general class of walks. ⇒ Counting results. ⇒ Random sampling of triangulations in linear time. 4 n � 3 n � k n = . ( n + 1)(2 n + 1) n Boston, March 2006 Olivier Bernardi - LaBRI – p.26/32

Open problems Is it possible to describe the conjugacy class of a Kreweras walk without using the cubic map ? 4 n � 3 n � k n = . ( n + 1)(2 n + 1) n Boston, March 2006 Olivier Bernardi - LaBRI – p.27/32

Open problems Can we count Kreweras walks ending at ( i, 0) ? at ( i, j ) ? Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.28/32

Open problems Can we count Kreweras walks ending at ( i, 0) ? at ( i, j ) ? Theorem [Kreweras 65] : � 2 i � 2 i + 1 � 3 n + 2 i � k n,i = 4 n . ( n + i + 1)(2 n + 2 i + 1) i n Boston, March 2006 Olivier Bernardi - LaBRI – p.28/32

Open problems Kreweras walks ending at ( i, 0) are in bijection with ( i + 2) -near-cubic maps + depth-trees. Corollary: k n,i = 2 n × c n,i . Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.29/32

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

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

Open problems Can we extend this bijection to some other class of maps ? To quadrangulations ? Boston, March 2006 Olivier Bernardi - LaBRI – p.30/32

Open problems There are similar counting results: - Non-separable maps [Tutte] . - Two-stack sortable permutations [West, Zeilberger] . � 3 n � 2 NS n = . ( n + 1)(2 n + 1) n Boston, March 2006 ▽ Olivier Bernardi - LaBRI – p.31/32