## Bijections for Baxter Families and Related Objects (2008)

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Citations: | 7 - 5 self |

### BibTeX

@MISC{Felsner08bijectionsfor,

author = {Stefan Felsner and Éric Fusy and Marc Noy and David Orden},

title = {Bijections for Baxter Families and Related Objects},

year = {2008}

}

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### Abstract

The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)

### Citations

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Citation Context ...th n + 1 leaves, as well as plane trees with n + 1 vertices, are counted by the Catalan number Cn = 1 ( ) 2n n+1 n . Catalan numbers are found in The On-Line Encyclopedia of Catalan Integer Sequences =-=[41]-=- as sequence A000108. Stanley [42, Exercise 6.19] collected 66 Catalan number families. Although the subject is well-studied, we include a particular proof showing that binary trees are a Catalan fami... |

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Citation Context ...case of separating decompositions we also found the tree decomposition being implied by local conditions (see item (3) after Theorem 2.5). Note. Schnyder woods were introduced by Schnyder in [39] and =-=[40]-=-. They have numerous applications in the context of graph drawing, e.g., [3, 7, 32], dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar structures... |

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Citation Context ... where Pβ is from (0, 1) to (ℓ, k + 1) and Pα is from (1, 0) to (ℓ + 1, k). The advantage of working with non-intersecting lattice paths is that now we can apply the Lemma of Lindström Gessel-Viennot =-=[27]-=-. Theorem 4.6. The number of binary trees with k + 1 left leaves and ℓ + 1 right leaves is (( ) ( )) k+ℓ k+ℓ ( )( ) k k−1 1 k + ℓ + 1 k + ℓ + 1 det ) ( ) k+ℓ = k + ℓ + 1 k k + 1 ( k+ℓ k+1 k This is th... |

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Citation Context ...ne bipolar orientation (this statement holds more generally with “graphs” instead of “maps”). Note. Plane bipolar orientations yield geometric representations of graphs in various flavors (visibility =-=[43]-=-, floor planning [37, 30], straight-line drawing [44, 23]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [17]; see also [16] for a detailed survey.... |

80 | The number of Baxter permutations - Chung, Graham, et al. - 1978 |

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Citation Context ...n (this statement holds more generally with “graphs” instead of “maps”). Note. Plane bipolar orientations yield geometric representations of graphs in various flavors (visibility [43], floor planning =-=[37, 30]-=-, straight-line drawing [44, 23]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [17]; see also [16] for a detailed survey. A map is bipartite if it... |

58 |
Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems, Theor
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Citation Context ...n (this statement holds more generally with “graphs” instead of “maps”). Note. Plane bipolar orientations yield geometric representations of graphs in various flavors (visibility [43], floor planning =-=[37, 30]-=-, straight-line drawing [44, 23]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [17]; see also [16] for a detailed survey. A map is bipartite if it... |

53 |
Planar graphs and poset dimension
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Citation Context ...t in the case of separating decompositions we also found the tree decomposition being implied by local conditions (see item (3) after Theorem 2.5). Note. Schnyder woods were introduced by Schnyder in =-=[39]-=- and [40]. They have numerous applications in the context of graph drawing, e.g., [3, 7, 32], dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar s... |

50 |
Planar grid embedding in linear time
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Citation Context ...erally with “graphs” instead of “maps”). Note. Plane bipolar orientations yield geometric representations of graphs in various flavors (visibility [43], floor planning [37, 30], straight-line drawing =-=[44, 23]-=-). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [17]; see also [16] for a detailed survey. A map is bipartite if its vertices can be partitioned in... |

45 | Expansive motions and the polytope of pointed pseudo-triangulations
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Citation Context ...endez and Pach. General planar graphs may require as many as four pages for a book embedding, Yannakakis [45]. Alternating layouts of plane trees in our sense were studied by Rote, Streinu and Santos =-=[38]-=- as non-crossing alternating trees. Gelfand et al. [26] call this class of trees standard trees; they show that these trees are a Catalan family. In [38] connections with rigidity theory and the geome... |

37 | Optimal coding and sampling of triangulations - Poulalhon, Schaeffer |

31 |
Embedding planar graphs in four pages
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Citation Context ...iginal source for a 2-book embedding of a quadrangulation is [15], by de Fraysseix, Ossona de Mendez and Pach. General planar graphs may require as many as four pages for a book embedding, Yannakakis =-=[45]-=-. Alternating layouts of plane trees in our sense were studied by Rote, Streinu and Santos [38] as non-crossing alternating trees. Gelfand et al. [26] call this class of trees standard trees; they sho... |

22 |
Orientations bipolaires
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Citation Context ...graphs in various flavors (visibility [43], floor planning [37, 30], straight-line drawing [44, 23]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications =-=[17]-=-; see also [16] for a detailed survey. A map is bipartite if its vertices can be partitioned into black and white vertices such that every edge connects a black vertex with a white vertex. Bipartite p... |

21 | Combinatorics of hypergeometric functions associated with positive roots
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- 1997
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Citation Context ...ny as four pages for a book embedding, Yannakakis [45]. Alternating layouts of plane trees in our sense were studied by Rote, Streinu and Santos [38] as non-crossing alternating trees. Gelfand et al. =-=[26]-=- call this class of trees standard trees; they show that these trees are a Catalan family. In [38] connections with rigidity theory and the geometry of the associahedron are established. As we have se... |

20 | The order dimension of convex polytopes - Brightwell, Trotter - 1993 |

20 | A left-first search algorithm for planar graphs. Discrete Computational Geometry 13:459–468
- Fraysseix, Mendez, et al.
- 1995
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Citation Context ...d consists in ranking each vertex v on the spine of the 2-book embedding according to the number of faces in a specific region R(v). The original source for a 2-book embedding of a quadrangulation is =-=[15]-=-, by de Fraysseix, Ossona de Mendez and Pach. General planar graphs may require as many as four pages for a book embedding, Yannakakis [45]. Alternating layouts of plane trees in our sense were studie... |

19 | Dissections and trees, with applications to optimal mesh encoding and to random sampling - Fusy, Poulalhon, et al. - 2005 |

18 | Strictly convex drawings of planar graphs
- Rote
- 2005
(Show Context)
Citation Context ...implied by local conditions (see item (3) after Theorem 2.5). Note. Schnyder woods were introduced by Schnyder in [39] and [40]. They have numerous applications in the context of graph drawing, e.g., =-=[3, 7, 32]-=-, dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar structures, e.g., [36, 24]. The connection with 3-orientations was found by de Fraysseix and ... |

18 |
A bijection between realizers of maximal plane graphs and pairs of noncrossing Dyck paths
- Bonichon
- 2005
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Citation Context ...ollows that the orientation and coloring of edges is a separating decomposition. In Section 8 we use this and some previous bijections to give an independent proof for a beautiful formula of Bonichon =-=[6]-=- for the number of Schnyder woods on triangulations with n vertices. 3 From Separating Decompositions to Twin Pairs of Binary Trees An embedding of a graph is called a 2-book embedding if the vertices... |

17 | A course in enumeration - Aigner - 2007 |

17 |
Optimal coding and sampling of triangulations. Algorithmica
- Poulalhon, Schaeffer
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Citation Context ...ave numerous applications in the context of graph drawing, e.g., [3, 7, 32], dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar structures, e.g., =-=[36, 24]-=-. The connection with 3-orientations was found by de Fraysseix and Ossona de Mendez [14]. The aim of this section is to prove the following theorem of Bonichon. 27Theorem 8.4 (Bonichon [6]). The tota... |

16 |
Bipolar orientations revisited
- Fraysseix, Mendez, et al.
- 1995
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Citation Context ...us flavors (visibility [43], floor planning [37, 30], straight-line drawing [44, 23]). The thesis of Ossona de Mendez is devoted to studying their beautiful properties and applications [17]; see also =-=[16]-=- for a detailed survey. A map is bipartite if its vertices can be partitioned into black and white vertices such that every edge connects a black vertex with a white vertex. Bipartite plane graphs are... |

14 |
On grid intersection graphs
- Hartman, Newman, et al.
- 1991
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Citation Context ...ry trees. The following theorem follows from the above discussion. Theorem 5.4. There is a bijection between twin pairs of binary trees with n leaves and rectangulations of Xn−2. Note. Hartman et al. =-=[29]-=- and later independently de Fraysseix et al. [15] prove that it is possible to assign a set of internally disjoint vertical and horizontal segments to the vertices of any bipartite map G such that two... |

13 | Dichromatic polynomials and Potts models summed over rooted maps - Baxter - 2013 |

11 | Binary labelings for plane quadrangulations and their
- Felsner, Huemer, et al.
- 2010
(Show Context)
Citation Context ...tly by induction on the height of each tree, using the local rules in Definition 2.2. See also Figure 4 for an example. Note. A proof of Proposition 3.3 was given by Felsner, Huemer, Kappes and Orden =-=[20]-=-. These authors study what they call strong binary labelings of the angles of a quadrangulation. They show that these labelings are in bijection with 2-orientations and separating decompositions. In t... |

11 |
Intervals in Catalan lattices and realizers of triangulations
- Bernardi, Bonichon
(Show Context)
Citation Context ... to be equal and the remaining pair is a non-crossing pair of Dyck paths. This implies the formula. Note. The original proof, Bonichon [6], and a more recent simplified version, Bernardi and Bonichon =-=[5]-=-, are also based on a bijection between Schnyder woods and pairs of non-crossing Dyck paths. In [5] the authors also enumerate special classes of Schnyder woods. Fusy, Schaeffer and Poulalhon have a b... |

10 |
Convex drawings of 3-connected planar graphs
- Bonichon, Felsner, et al.
(Show Context)
Citation Context ...implied by local conditions (see item (3) after Theorem 2.5). Note. Schnyder woods were introduced by Schnyder in [39] and [40]. They have numerous applications in the context of graph drawing, e.g., =-=[3, 7, 32]-=-, dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar structures, e.g., [36, 24]. The connection with 3-orientations was found by de Fraysseix and ... |

10 | On topological aspects of orientations
- Fraysseix, Mendez
(Show Context)
Citation Context ...tation if every 2vertex, except s and t, has outdegree two. orientation Double-counting the edges of Q ensures that s and t are sinks in every 2-orientation. 5Theorem 2.5 (de Fraysseix and de Mendez =-=[14]-=-). Separating decompositions and 2-orientations of a quadrangulation Q are in bijection. From the proof we obtain an additional property of a separating decomposition: (3) The red edges form a tree di... |

10 |
Stack words, standard tableaux and Baxter permutations, Discrete Math
- Dulucq, Guibert
- 1996
(Show Context)
Citation Context ...yana [35]. Stanley [42, Exercise 6.36] recommends to look for decompositions into subsets counted by Narayana numbers in Catalan families. The notion of fingerprints is implicit in Dulucq and Guibert =-=[18]-=-. The combination with the bodyprint and the related bijections are new, according to our knowledge. 5 Four Incarnations of Twin Pairs of Trees After the Catalan and Narayana digression we now come ba... |

9 |
permutations rise again
- Baxter
- 1979
(Show Context)
Citation Context ...mbers first appeared in the context of counting Baxter permutations. Chung, Graham, Hoggatt and Kleiman [12] found some interesting recurrences and gave a proof based on generating functions. Mallows =-=[34]-=- found the refined count of Baxter permutations by rises (Proposition 6.8). The bijection of Theorem 6.4 is essentially due to Dulucq and Guibert [18, 19]. Their description and proof, however, does n... |

7 | permutations - Dulucq, Guibert, et al. - 1998 |

6 | Improved compact visibility representation of planar graphs via Schnyder’s realizer
- Lin, Lu, et al.
(Show Context)
Citation Context ...implied by local conditions (see item (3) after Theorem 2.5). Note. Schnyder woods were introduced by Schnyder in [39] and [40]. They have numerous applications in the context of graph drawing, e.g., =-=[3, 7, 32]-=-, dimension theory for orders, graphs and polytopes, e.g., [39, 9, 21], enumeration and encoding of planar structures, e.g., [36, 24]. The connection with 3-orientations was found by de Fraysseix and ... |

5 | Straight-line drawing of quadrangulations - Fusy - 2006 |

5 |
Doubly alternating Baxter permutations are
- Guibert, Linusson
(Show Context)
Citation Context ...mutations and shuffle of parenthesis systems. The theorem was reproved by Dulucq and Guibert [18] as a specialization of their bijection between Baxter permutations and twin pairs of binary trees. In =-=[28]-=- it is shown that alternating Baxter permutations with the property that their inverse is again alternating Baxter are counted by the Catalan numbers. In recent work of Bonichon, Bousquet-Mélou and Fu... |

4 | The complexities of the coefficients of the Tutte polynomial - Annan - 1995 |

4 |
The order dimension of convex polytopes
- Brightwell, Trotter
- 1993
(Show Context)
Citation Context ...chnyder woods were introduced by Schnyder in [39] and [40]. They have numerous applications in the context of graph drawing, e.g., [3, 7, 32], dimension theory for orders, graphs and polytopes, e.g., =-=[39, 9, 21]-=-, enumeration and encoding of planar structures, e.g., [36, 24]. The connection with 3-orientations was found by de Fraysseix and Ossona de Mendez [14]. The aim of this section is to prove the followi... |

4 |
les treillis formés par les partitions d’une unties et leurs applications à la théorie des probabilités
- Narayana, Sur
- 1955
(Show Context)
Citation Context ...e On-Line Encyclopedia of Integer Sequences [41] as sequence A001263. The Narayana numbers (actually, a q-analogue of them) were first studied by MacMahon [33] and were later rediscovered by Narayana =-=[35]-=-. Stanley [42, Exercise 6.36] recommends to look for decompositions into subsets counted by Narayana numbers in Catalan families. The notion of fingerprints is implicit in Dulucq and Guibert [18]. The... |

3 | On the Number of Planar Orientations with Prescribed Degrees
- Felsner, Zickfeld
(Show Context)
Citation Context ...r number. In contrast to the nice formulas for the number of 2-orientations of quadrangulations on n vertices, very little is known about the number of 2-orientations of a fixed quadrangulation Q. In =-=[22]-=- it is shown that the maximal number of 2-orientations a quadrangulation on n vertices can have is asymptotically between 1, 47 n and 1, 91 n . To our knowledge, the computational complexity of the co... |

2 | On the number of rectangulations of a planar point set
- Ackerman, Barequet, et al.
(Show Context)
Citation Context ...re a touching segment representation for Q. Deleting some and retracting the ends of some other segments yields a representation for G. A similar observation was made by Ackerman, Barequet and Pinter =-=[1]-=-. Next, we map twin pairs of binary trees to triples of non-intersecting lattice paths. The map is based on the fingerprint and bodyprint of a binary tree, and the lattice paths associated to them. 15... |

2 | Bijective counting of bipolar orientations - Fusy, Poulalhon, et al. - 2007 |

2 | Orthogonal surfaces and their cp-orders
- Felsner, Kappes
(Show Context)
Citation Context ...chnyder woods were introduced by Schnyder in [39] and [40]. They have numerous applications in the context of graph drawing, e.g., [3, 7, 32], dimension theory for orders, graphs and polytopes, e.g., =-=[39, 9, 21]-=-, enumeration and encoding of planar structures, e.g., [36, 24]. The connection with 3-orientations was found by de Fraysseix and Ossona de Mendez [14]. The aim of this section is to prove the followi... |

1 | Catalan intervals and realizers of triangulations. arXiv.org:0704.3731 - Bernardi, Bonichon |

1 |
Shuffle of pharentesis systems and Baxter permutations
- Cori, Dulucq, et al.
- 1986
(Show Context)
Citation Context ...se n = 2k tree with k leaves tree with k leaves tree with k leaves tree with k + 1 leaves Figure 18: Alternating Baxter permutations and pairs of trees. Note. Theorem 6.10 was obtained by Cori et al. =-=[13]-=-. They gave a nice bijection between alternating Baxter permutations and shuffle of parenthesis systems. The theorem was reproved by Dulucq and Guibert [18] as a specialization of their bijection betw... |

1 | High-dimensional orthogonal surfaces - Kappes - 2006 |

1 |
permutations and plane bipolar orientations
- Bonichon, Bousquet-Mélou, et al.
(Show Context)
Citation Context ... is shown that alternating Baxter permutations with the property that their inverse is again alternating Baxter are counted by the Catalan numbers. In recent work of Bonichon, Bousquet-Mélou and Fusy =-=[8]-=- a direct bijection between Baxter permutations and plane bipolar orientations was given. Their procedure consists in drawing segments in the diagrammatic representation of a Baxter permutation so as ... |

1 |
Straight-line drawing of quadrangulations, in Proceedings of Graph Drawing ’06, vol. 4372 of LNCS, 2007
- Fusy
(Show Context)
Citation Context ...to include some more refined counts. A very good entrance point for more information about Baxter numbers is The On-Line Encyclopedia of Integer Sequences [41, A001181]. Fusy, Schaeffer and Poulalhon =-=[25]-=- gave a direct bijection from separating decompositions to triples of non-intersecting paths in a grid. Similarly as in this article, they obtain the triple of paths from a triple of words encoding th... |

1 |
Combinatorial Analysis, Vols. 1 and 2, Camb
- MacMahon
- 1915
(Show Context)
Citation Context ...ℓ). Note. Narayana numbers can be found in The On-Line Encyclopedia of Integer Sequences [41] as sequence A001263. The Narayana numbers (actually, a q-analogue of them) were first studied by MacMahon =-=[33]-=- and were later rediscovered by Narayana [35]. Stanley [42, Exercise 6.36] recommends to look for decompositions into subsets counted by Narayana numbers in Catalan families. The notion of fingerprint... |