, Volume 38, Issue 2, pp 443–486 | Cite as

Associahedra Via Spines

  • Carsten LangeEmail author
  • Vincent Pilaud
Original Paper


An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange’s construction closer to J.-L. Loday’s original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.

Mathematics Subject Classification (2000)

52B05 52B11 20F55 


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  1. [1]
    F. Ardila, C. Benedetti and J. Doker: Matroid polytopes and their volumes, Discrete Comput. Geom. 43 (2010), 841–854.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. J. Billera, P. Filliman and B. Sturmfels: Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Björner and M. Wachs: Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349 (1997), 3945–3975.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. P. Carr and S. L. Devadoss: Coxeter complexes and graph-associahedra, Topology Appl. 153 (2006), 2155–2168.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Ceballos, J.-Ph. Labbé and C. Stump: Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin., 2013.Google Scholar
  6. [6]
    C. Ceballos, F. Santos and G. M. Ziegler: Many non-equivalent realizations of the associahedron, Combinatorica 35 (2015), 513–551.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Chapoton, S. Fomin and A. Zelevinsky: Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), 537–566.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    G. Chatel and V. Pilaud: Cambrian Hopf Algebras, Adv. Math. 311 (2017), 598–633.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein: Introduction to algorithms, MIT Press, Cambridge, MA, third edition, 2009.zbMATHGoogle Scholar
  10. [10]
    S. L. Devadoss: A realization of graph associahedra, Discrete Math. 309 (2009), 271–276.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Fomin and A. Zelevinsky: Cluster algebras I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. Fomin and A. Zelevinsky: Cluster algebras II. Finite type classification, Invent. Math. 154 (2003), 63–121.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Á. Galambos and V. Reiner: Acyclic sets of linear orders via the Bruhat orders, Soc. Choice Welfare 30 (2008), 245–264.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    I. Gelfand, M. Kapranov and A. Zelevinsky: Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2008, Reprint of the 1994 edition.zbMATHGoogle Scholar
  15. [15]
    C. Hohlweg: Permutahedra and associahedra, Pages 129-159 in [49].Google Scholar
  16. [16]
    C. Hohlweg and C. Lange: Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), 517–543.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    C. Hohlweg, C. Lange and H. Thomas: Permutahedra and generalized associahedra, Adv. Math. 226 (2011), 608–640.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C. Hohlweg, J. Lortie and A. Raymond: The centers of gravity of the associahedron and of the permutahedron are the same, Electron. J. Combin. 17 Research Paper 72, 14, 2010.zbMATHGoogle Scholar
  19. [19]
    K. Igusa and J. Ostroff: Mixed cobinary trees, preprint, arXiv:1307.3587, 2013.Google Scholar
  20. [20]
    A. Knutson and E. Miller: Subword complexes in Coxeter groups, Adv. Math. 184 (2004), 161–176.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.-Ph. Labbé and C. Lange: Cambrian acyclic domains: counting c-singletons, in preparation, 2017.Google Scholar
  22. [22]
    C. Lange: Minkowski decomposition of associahedra and related combinatorics, Discrete Comput. Geom. 50 (2013), 903–939.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    C. W. Lee: The associahedron and triangulations of the n-gon, European J. Combin. 10 (1989), 551–560.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J.-L. Loday: Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), 267–278.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J.-L. Loday and M. O. Ronco: Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293–309.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C. Malvenuto and C. Reutenauer: Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967–982.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. V. Narayana: Lattice path combinatorics with statistical applications, volume 23 of Mathematical Expositions, University of Toronto Press, Toronto, Ont., 1979.zbMATHGoogle Scholar
  28. [28]
    V. Pilaud: Signed tree associahedra, Extended abstract in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), preprint, arXiv:1309.5222, 2013.Google Scholar
  29. [29]
    V. Pilaud: Which nestohedra are removahedra?, Rev. Colomb. Math. 51 (2017), 21–42.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    V. Pilaud and M. Pocchiola: Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom. 48 (2012), 142–191.CrossRefzbMATHGoogle Scholar
  31. [31]
    V. Pilaud and F. Santos: The brick polytope of a sorting network, European J. Combin. 33 (2012), 632–662.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    V. Pilaud and C. Stump: EL-labelings and canonical spanning trees for subword complexes, in: Discrete Geometry and Optimization, Fields Institute Communications Series, 213–248. Springer, 2013.CrossRefGoogle Scholar
  33. [33]
    V. Pilaud and C. Stump: Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math. 276 (2015), 1–61.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    V. Pilaud and C. Stump: Vertex barycenter of generalized associahedra, Proc. Amer. Math. Soc. 143 (2015), 2623–2636.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A. Postnikov: Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN 6 (2009), 1026–1106.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Postnikov, V. Reiner and L. K. Williams: Faces of generalized permutohedra, Doc. Math. 13 (2008), 207–273.MathSciNetzbMATHGoogle Scholar
  37. [37]
    L. Pournin: The diameter of associahedra, Adv. Math. 259 (2014), 13–42.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N. Reading: Lattice congruences of the weak order, Order 21 (2004), 315–344.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    N. Reading: Cambrian lattices, Adv. Math. 205 (2006), 313–353.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    N. Reading and D. E. Speyer: Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), 407–447.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    S. Shnider and S. Sternberg: Quantum groups: From coalgebras to Drinfeld algebras, Series in Mathematical Physics. International Press, Cambridge, MA, 1993.zbMATHGoogle Scholar
  42. [42]
    D. D. Sleator, R. E. Tarjan and W. P. Thurston: Rotation distance, triangulations, and hyperbolic space, J. Amer. Math. Soc. 1 (1988), 647–681.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    L. Solomon: A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255–264.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    R. P. Stanley: Hipparchus, Plutarch, Schröder, and Hough, Amer. Math. Monthly 104 (1997), 344–350.zbMATHGoogle Scholar
  45. [45]
    R. P. Stanley: Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.CrossRefzbMATHGoogle Scholar
  46. [46]
    J. Stasheff: Homotopy associativity of H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312.MathSciNetzbMATHGoogle Scholar
  47. [47]
    J. Stasheff: From operads to ☎ysically" inspired theories, in: Operads: Proceedings of Renaissance Conferences (Hartfort, CT/Luminy, 1995), volume 202 of Contemporary Mathematics, 53–81, Cambridge, MA, 1997. American Mathematical Society, Appendix B by Steve Shnider and Jim Stasheff for a corrected polytope construction.CrossRefGoogle Scholar
  48. [48]
    S. Stella: Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), 121–158.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    Associahedra, Tamari Lattices and Related Structures, Tamari Memorial Festschrift, F. Müller-Hoissen, J. M. Pallo and J. Stasheff eds., volume 299 of Progress in Mathematics, Springer, New York, 2012.Google Scholar
  50. [50]
    X. Viennot: Catalan tableaux and the asymmetric exclusion process, in: 19th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2007). 2007.Google Scholar
  51. [51]
    A. Zelevinsky: Nested complexes and their polyhedral realizations, Pure Appl. Math. Q. 2 (2006), 655–671.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität MünchenMünchenGermany
  2. 2.Freie Universität BerlinBerlinGermany
  3. 3.CNRS & École PolytechniqueParisFrance

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