SpringerLink
Log in

Geometric Realizations of Tamari Interval Lattices Via Cubic Coordinates

  • Published:
Order Aims and scope Submit manuscript

Abstract

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernardi, O., Bonichon, N.: Intervals in Catalan lattices and realizers of triangulations. J. Combin. Theory Ser. A 116(1), 55–75 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bousquet-Mélou, M., Fusy, É., Préville-Ratelle, L.-F.: The number of intervals in the m-Tamari lattices. Electronic J. Combin., 18(2) (2012)

  3. Bergeron, F., Préville-Ratelle, L.-F.: Higher trivariate diagonal harmonics via generalized Tamari posets. J. Combin. (3):317–341 (2012)

  4. Björner, A., Wachs, M.L.: Shellable nonpure complexes and posets. I. Trans. Amer. Math. Soc. 348(4), 1299–1327 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björner, A., Wachs, M. L.: Shellable nonpure complexes and posets. II. Trans. Amer. Math. Soc. 349(10), 3945–3975 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chapoton, F.: Sur le nombre d’intervalles dans les treillis de Tamari. Sém. Lothar. Combin., 55:Art, B55f, 18 (2006)

  7. Chapoton, F.: Une note sur les intervalles de Tamari. Ann. Math. Blaise Pascal 25(2), 299–314 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Combe, C.: Cubic realizations of Tamari interval lattices. Sém. Lothar. Combin. 82B.23, 12 pp (2019)

    MATH  Google Scholar 

  9. Châtel, G., Pons, V.: Counting smaller elements in the Tamari and m-Tamari lattices. J. Combin. Theory Ser. A 134, 58–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fang, W.: Planar triangulations, bridgeless planar maps and and Tamari intervals. European J. Combin. 70, 75–91 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fang, W., Préville-Ratelle, L.-F.: The enumeration of generalized Tamari intervals. European J. Combin. 61, 69–84 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giraudo, S.: Algebraic and combinatorial structures on pairs of twin binary trees. J Algebra 360, 115–157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, S., Tamari, D.: Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law. J. Combinatorial Theory Ser. A 13, 7–13 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Loday, J.-L.: The Diagonal of the Stasheff polytope. In: Higher structures in Geometry and Physics, vol. 287 of Progr. Math.269-292, pp. 269–292. Birkhäuser/Springer, New York (2011)

  15. Milner, E. C., Pouzet, M.: A note on the dimension of a poset. Order 7(1), 101–102 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Markl, M., Shnider, S.: Associahedra, cellular W-construction and products of \(A_{\infty }\)-algebras. Trans. Amer. Math. Soc. 358(6), 2353–2372 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Masuda, N., Thomas, H., Tonks, A., Vallette, B.: The diagonal of the associahedra. J. É,c. Polytech. Math. 8, 121–146 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pallo, J.M.: Enumerating, ranking and unranking binary trees. Comput. J. 29(2), 171–175 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Préville-Ratelle, L.-F., Viennot, X.: The enumeration of generalized Tamari intervals. Trans. Amer. Math. Soc. 369(7), 5219–5239 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rognerud, B.: Exceptional and modern intervals of the Tamari lattice. Sém. Lothar. Combin., 79:Art. B79d, 23, 2018-2020

  21. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org/

  22. Stanley, R. P.: Enumerative Combinatorics. Volume 1, Volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press. Cambridge, second edition (2012)

  23. Saneblidze, S., Umble, R.: Diagonals on the permutahedra, multiplihedra and associahedra. Homology Homotopy Appl. 6(1), 363–411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tamari, D.: The algebra of bracketings and their enumeration. Nieuw Arch. Wisk. 10(3), 131–146 (1962)

    MathSciNet  MATH  Google Scholar 

  25. Trotter, W.T.: Combinatorics and partially ordered sets: dimension theory. Johns Hopkins Series in the Mathematical Sciences, The Johns Hopkins University Press (2002)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous reviewer for all his good advices, which contributed to the improvement of this article. The author would also like to thank Frédéric Chapoton, Samuele Giraudo, and Baptiste Rognerud for the numerous discussions and their suggestions.

My manuscript has no associated data.

Author information

Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Camille Combe

Authors
  1. Camille Combe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Camille Combe.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Combe, C. Geometric Realizations of Tamari Interval Lattices Via Cubic Coordinates. Order 40, 589–621 (2023). https://doi.org/10.1007/s11083-023-09624-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-023-09624-y

Keywords

Search

Navigation