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.

Received: 2018-10-11
Revised: 2019-09-20
Accepted: 2019-09-20
Published online: 2020-04-01
DOI: https://doi.org/10.5802/alco.98
Classification: 05E,  05A15,  05A19,  06A07,  52B
Keywords: Dyck path, semilattice, enumerative combinatorics, interval, Hochschild polytope
@article{ALCO_2020__3_2_433_0,
     author = {Chapoton, Fr\'ed\'eric},
     title = {Some properties of a new partial order on~Dyck paths},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     pages = {433-463},
     doi = {10.5802/alco.98},
     language = {en},
     url={alco.centre-mersenne.org/item/ALCO_2020__3_2_433_0/}
}
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/item/ALCO_2020__3_2_433_0/

[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 | Article | MR 3029440 | Zbl 1291.05213

[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 | Article | MR 2469248 | Zbl 1161.06001

[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, P31, 26 pages | MR 2880681 | Zbl 1262.05005

[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 | Article | MR 2236503 | Zbl 1099.05043

[5] Chapoton, Frédéric Sur le nombre d’intervalles dans les treillis de Tamari, Sémin. Lothar. Comb., Volume 55 (2005/07), Art. B55f, 18 pages | MR 2264942 | Zbl 1207.05011

[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 | Article | MR 3221542 | Zbl 1264.16014

[7] Fang, Wenjie Planar triangulations, bridgeless planar maps and Tamari intervals, Eur. J. Comb., Volume 70 (2018), pp. 75-91 | Article | MR 3779605 | Zbl 1384.05072

[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 | Article | MR 3759209

[9] Fang, Wenjie; Préville-Ratelle, Louis-François The enumeration of generalized Tamari intervals, Eur. J. Comb., Volume 61 (2017), pp. 69-84 | Article | MR 3588709 | Zbl 1352.05191

[10] Freese, Ralph; Ježek, Jaroslav; Nation, James B. Free lattices, Math. Surv. Monogr., Volume 42, American Mathematical Society, Providence, RI, 1995, viii+293 pages | Article | MR 1319815 | Zbl 0839.06005

[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 | Article | MR 0238984 | Zbl 0158.01904

[12] Grätzer, George Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel, 2011, xxx+613 pages | Article | MR 2768581 | Zbl 1233.06001

[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 | Article | MR 1714618 | Zbl 0941.16007

[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 | Article | MR 2762549 | Zbl 1220.18007

[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., Volume 299, Birkhäuser/Springer, Basel, 2012, xx+433 pages (Tamari memorial Festschrift) | Article | MR 3235205 | Zbl 1253.00013

[18] Pallo, Jean Marcel Right-arm rotation distance between binary trees, Inf. Process. Lett., Volume 87 (2003) no. 4, pp. 173-177 | Article | MR 1994347 | Zbl 1161.68692

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

[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 | Article | MR 2498922 | Zbl 1181.55009

[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 3385336 | Zbl 1191.55011

[23] Tutte, William T. A census of planar maps, Can. J. Math., Volume 15 (1963), pp. 249-271 | Article | MR 0146823 | Zbl 0115.17305