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### A bijective proof of an identity for noncrossing graphs

Pavel Podbrdský
2003 Discrete Mathematics
We give a bijective proof for the identity an+2 = 8bn, where an is the number of noncrossing simple graphs with n (possibly isolated) vertices and bn is the number of noncrossing graphs without isolated  ...  Klazar for bringing this problem to my attention when I attended the course "Combinatorial seminar" in the spring semester, 2000.  ...  Comparision of both expressions for a n and b n gives the identity. Our aim in the present note is to give a bijective proof without use of generating functions. Theorem 1.  ...

### Descents in Noncrossing Trees

David S. Hough
2003 Electronic Journal of Combinatorics
A bijection shows combinatorially why the descent generating function with descents set equal to $2$ is the generating function for connected noncrossing graphs.  ...  The generating function for descents in noncrossing trees is found.  ...  D(z, u) evaluated at 2 is the g.f. for connected noncrossing graphs. A bijection between noncrossing trees weighted by descents and connected noncrossing graphs Proof.  ...

### Noncrossing Trees and Noncrossing Graphs

William Y. C. Chen, Sherry H. F. Yan
2006 Electronic Journal of Combinatorics
We use the representation of Panholzer and Prodinger for noncrossing trees and find a correspondence between a class of noncrossing trees, called proper noncrossing trees, and the set of symmetric ternary  ...  The second result of this paper is a parity reversing involution on connected noncrossing graphs which leads to a relation between the number of noncrossing trees with $n$ edges and $k$ descents and the  ...  We would like to thank the referee for helpful suggestions.  ...

### Noncrossing Trees and Noncrossing Graphs [article]

William Y.C. Chen, Sherry H.F. Yan
2005 arXiv   pre-print
and the number of connected noncrossing graphs with a given number of vertices and edges.  ...  We use the representation of Panholzer and Prodinger for noncrossing trees and find a correspondence between a class of noncrossing trees, called proper oncrossing trees, and the set of symmetric ternary  ...  This work was supported by the 973 Project on Mathematical Mechanization, the National Science Foundation, the Ministry of Education, and the Ministry of Science and Technology of China.  ...

### On trees and noncrossing partitions

Martin Klazar
1998 Discrete Applied Mathematics
We give a simple and natural proof of (an extension of) the identity P(X. 1. )I ) = t-'2( X I. I -I. II --I ). The number P(li, I, 17) counts noncrossing partitions of { I, 2.  ...  The lower index 2 indicate\ partitions with no part of size three or more. We USC the identity to give quick proofs of the closed t'ormulac for P(k.  ...  For a bijective proof of (5) using bracketings see [9] .  ...

### Reduction of m-regular noncrossing partitions

William Y.C. Chen, Eva Y.P. Deng, Rosena R.X. Du
2005 European journal of combinatorics (Print)
For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which  ...  This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures.  ...  Zeng for helpful comments. We also thank the referees for important suggestions adopted in the revised version.  ...

### Reduction of m-Regular Noncrossing Partitions [article]

William Y. C. Chen, Eva Y. P. Deng, Rosena R. X. Du
2004 arXiv   pre-print
For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which  ...  This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures.  ...  Zeng for helpful comments. We also thank the referees for important suggestions adopted in the revised version.  ...

### k-noncrossing and k-nonnesting graphs and fillings of Ferrers diagrams [article]

Anna de Mier
2006 arXiv   pre-print
We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence.  ...  In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling.  ...  Acknowledgements I am very grateful to Sergi Elizalde, Vít Jelínek, Martin Klazar, Martin Loebl, and Marc Noy for many fruitful and stimulating discussions and for pointing out several useful references  ...

### Undesired parking spaces and contractible pieces of the noncrossing partition link [article]

Michael Dougherty, Jon McCammond
2017 arXiv   pre-print
The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former.  ...  In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.  ...  There is a natural bijection between noncrossing hypertrees in an n-gon and the even-sided dissections of a 2n-gon. Theorem 3.4 (Noncrossing hypertrees and polygon dissections).  ...

### Undesired Parking Spaces and Contractible Pieces of the Noncrossing Partition Link

Michael Dougherty, Jon McCammond
2018 Electronic Journal of Combinatorics
The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former.  ...  In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.  ...  Proof. For k = n, this follows from Theorem 5.5. For 1 < k < n, let = n − k > 0.  ...

### k-noncrossing and k-nonnesting graphs and fillings of ferrers diagrams

Anna de Mier
2007 Combinatorica
We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence.  ...  In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling.  ...  Acknowledgements I am very grateful to Sergi Elizalde, Vít Jelínek, Martin Klazar, Martin Loebl, and Marc Noy for many fruitful and stimulating discussions on the topics of this paper, and for pointing  ...

### Bijections between noncrossing and nonnesting partitions for classical reflection groups

Alex Fink, Benjamin Iriarte Giraldo
2010 Portugaliae Mathematica
We present type preserving bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner.  ...  To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all  ...  Our contribution has been to provide a family of bijective proofs, one for each type of classical reflection group, that also address equidistribution by type.  ...

### Bijections between noncrossing and nonnesting partitions for classical reflection groups

Alex Fink, Benjamin Iriarte Giraldo
2009 Discrete Mathematics & Theoretical Computer Science
International audience We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner  ...  To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all  ...  Our contribution has been to provide a family of bijective proofs, one for each type of classical reflection group, that also address equidistribution by type.  ...

### Maximal Chains in Bond Lattices

Shreya Ahirwar, Susanna Fishel, Parikshita Gya, Pamela Harris, Nguyen Pham, Andrés Vindas Meléndez, Dan Khanh Vo
2022 Electronic Journal of Combinatorics
Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice.  ...  We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices.  ...  Acknowledgements We would like to thank the referees for their suggestions, which we used to improve the paper.  ...

### Noncrossing partitions, Bruhat order and the cluster complex [article]

Philippe Biane, Matthieu Josuat-Vergès
2018 arXiv   pre-print
In particular we study the restriction of these orders to noncrossing partitions and show that the intervals for these orders can be enumerated in terms of the cluster complex.  ...  The properties of our orders permit to revisit several results in Coxeter combinatorics, such as the Chapoton triangles and how they are related, the enumeration of reflections with full support, the bijections  ...  Acknowledgement We thank Cesar Ceballos for explanations about the subword complex.  ...
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