Some properties of a new partial order on Dyck paths
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 433-463.

A new partial order is defined on the set of Dyck paths of a given length. This partial order is proved to be a meet-semilattice. Its intervals are enumerated and a specific interval is connected with an existing polytope coming from algebraic topology.

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DOI: 10.5802/alco.98
Classification: 05E, 05A15, 05A19, 06A07, 52B
Keywords: Dyck path, semilattice, enumerative combinatorics, interval, Hochschild polytope

Chapoton, Frédéric 1

1 Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg et CNRS 7 rue René Descartes 67000 Strasbourg, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Chapoton, Frédéric. Some properties of a new partial order on Dyck paths. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 433-463. doi : 10.5802/alco.98. https://alco.centre-mersenne.org/articles/10.5802/alco.98/

[1] Bergeron, François; Préville-Ratelle, Louis-François Higher trivariate diagonal harmonics via generalized Tamari posets, J. Comb., Volume 3 (2012) no. 3, pp. 317-341 | DOI | MR | Zbl

[2] Bernardi, Olivier; Bonichon, Nicolas Intervals in Catalan lattices and realizers of triangulations, J. Comb. Theory, Ser. A, Volume 116 (2009) no. 1, pp. 55-75 | DOI | MR | Zbl

[3] Bousquet-Mélou, Mireille; Fusy, Éric; Préville-Ratelle, Louis-François The number of intervals in the m-Tamari lattices, Electron. J. Comb., Volume 18 (2011) no. 2, Paper no. P31, 26 pages | MR | Zbl

[4] Bousquet-Mélou, Mireille; Jehanne, Arnaud Polynomial equations with one catalytic variable, algebraic series and map enumeration, J. Comb. Theory, Ser. B, Volume 96 (2006) no. 5, pp. 623-672 | DOI | MR | Zbl

[5] Chapoton, Frédéric Sur le nombre d’intervalles dans les treillis de Tamari, Sémin. Lothar. Comb., Volume 55 (2005/07), Paper no. Art. B55f, 18 pages | MR | Zbl

[6] Chapoton, Frédéric On the categories of modules over the Tamari posets, Associahedra, Tamari lattices and related structures (Prog. Math.), Volume 299, Birkhäuser/Springer, Basel, 2012, pp. 269-280 | DOI | MR | Zbl

[7] Fang, Wenjie Planar triangulations, bridgeless planar maps and Tamari intervals, Eur. J. Comb., Volume 70 (2018), pp. 75-91 | DOI | MR | Zbl

[8] Fang, Wenjie A trinity of duality: non-separable planar maps, β(1,0)-trees and synchronized intervals, Adv. Appl. Math., Volume 95 (2018), pp. 1-30 | DOI | MR | Zbl

[9] Fang, Wenjie; Préville-Ratelle, Louis-François The enumeration of generalized Tamari intervals, Eur. J. Comb., Volume 61 (2017), pp. 69-84 | DOI | MR | Zbl

[10] Freese, Ralph; Ježek, Jaroslav; Nation, James B. Free lattices, Math. Surv. Monogr., 42, American Mathematical Society, Providence, RI, 1995, viii+293 pages | DOI | MR | Zbl

[11] Friedman, Haya; Tamari, Dov Problèmes d’associativité: Une structure de treillis finis induite par une loi demi-associative, J. Comb. Theory, Volume 2 (1967), pp. 215-242 | DOI | MR | Zbl

[12] Grätzer, George Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel, 2011, xxx+613 pages | DOI | MR | Zbl

[13] Ladkani, Sefi Universal derived equivalences of posets of tilting modules (2007) (https://arxiv.org/abs/0708.1287)

[14] Lenzing, Helmut Coxeter transformations associated with finite-dimensional algebras, Computational methods for representations of groups and algebras (Essen, 1997) (Prog. Math.), Volume 173, Birkhäuser, Basel, 1999, pp. 287-308 | DOI | MR | Zbl

[15] Loday, Jean-Louis The diagonal of the Stasheff polytope, Higher structures in geometry and physics (Prog. Math.), Volume 287, Birkhäuser/Springer, New York, 2011, pp. 269-292 | DOI | MR | Zbl

[16] Masuda, Naruki; Thomas, Hugh; Tonks, Andy; Vallette, Bruno The diagonal of the associahedra (2019) (https://arxiv.org/abs/1902.08059)

[17] Associahedra, Tamari lattices and related structures (Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds.), Prog. Math., 299, Birkhäuser/Springer, Basel, 2012, xx+433 pages (Tamari memorial Festschrift) | DOI | MR | Zbl

[18] Pallo, Jean Marcel Right-arm rotation distance between binary trees, Inf. Process. Lett., Volume 87 (2003) no. 4, pp. 173-177 | DOI | MR | Zbl

[19] Rivera, Manuel; Saneblidze, Samson A combinatorial model for the free loop fibration (2017) (https://arxiv.org/abs/1712.02644) | Zbl

[20] Rognerud, Baptiste Exceptional and modern intervals of the Tamari lattice (2018) (to appear in Sémin. Lothar. Comb.)

[21] Saneblidze, Samson The bitwisted Cartesian model for the free loop fibration, Topology Appl., Volume 156 (2009) no. 5, pp. 897-910 | DOI | MR | Zbl

[22] Saneblidze, Samson On the homology theory of the closed geodesic problem, Rep. Enlarged Sess. Semin. I. Vekua Inst. Appl. Math., Volume 25 (2011), pp. 113-116 | MR

[23] Tutte, William T. A census of planar maps, Can. J. Math., Volume 15 (1963), pp. 249-271 | DOI | MR | Zbl

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