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.
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F. Ardila, C. Benedetti and J. Doker: Matroid polytopes and their volumes, Discrete Comput. Geom. 43 (2010), 841–854.
L. J. Billera, P. Filliman and B. Sturmfels: Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.
A. Björner and M. Wachs: Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349 (1997), 3945–3975.
M. P. Carr and S. L. Devadoss: Coxeter complexes and graph-associahedra, Topology Appl. 153 (2006), 2155–2168.
C. Ceballos, J.-Ph. Labbé and C. Stump: Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin., 2013.
C. Ceballos, F. Santos and G. M. Ziegler: Many non-equivalent realizations of the associahedron, Combinatorica 35 (2015), 513–551.
F. Chapoton, S. Fomin and A. Zelevinsky: Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), 537–566.
G. Chatel and V. Pilaud: Cambrian Hopf Algebras, Adv. Math. 311 (2017), 598–633.
T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein: Introduction to algorithms, MIT Press, Cambridge, MA, third edition, 2009.
S. L. Devadoss: A realization of graph associahedra, Discrete Math. 309 (2009), 271–276.
S. Fomin and A. Zelevinsky: Cluster algebras I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.
S. Fomin and A. Zelevinsky: Cluster algebras II. Finite type classification, Invent. Math. 154 (2003), 63–121.
Á. Galambos and V. Reiner: Acyclic sets of linear orders via the Bruhat orders, Soc. Choice Welfare 30 (2008), 245–264.
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.
C. Hohlweg: Permutahedra and associahedra, Pages 129-159 in .
C. Hohlweg and C. Lange: Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), 517–543.
C. Hohlweg, C. Lange and H. Thomas: Permutahedra and generalized associahedra, Adv. Math. 226 (2011), 608–640.
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.
K. Igusa and J. Ostroff: Mixed cobinary trees, preprint, arXiv:1307.3587, 2013.
A. Knutson and E. Miller: Subword complexes in Coxeter groups, Adv. Math. 184 (2004), 161–176.
J.-Ph. Labbé and C. Lange: Cambrian acyclic domains: counting c-singletons, in preparation, 2017.
C. Lange: Minkowski decomposition of associahedra and related combinatorics, Discrete Comput. Geom. 50 (2013), 903–939.
C. W. Lee: The associahedron and triangulations of the n-gon, European J. Combin. 10 (1989), 551–560.
J.-L. Loday: Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), 267–278.
J.-L. Loday and M. O. Ronco: Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), 293–309.
C. Malvenuto and C. Reutenauer: Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967–982.
T. V. Narayana: Lattice path combinatorics with statistical applications, volume 23 of Mathematical Expositions, University of Toronto Press, Toronto, Ont., 1979.
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.
V. Pilaud: Which nestohedra are removahedra?, Rev. Colomb. Math. 51 (2017), 21–42.
V. Pilaud and M. Pocchiola: Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom. 48 (2012), 142–191.
V. Pilaud and F. Santos: The brick polytope of a sorting network, European J. Combin. 33 (2012), 632–662.
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.
V. Pilaud and C. Stump: Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math. 276 (2015), 1–61.
V. Pilaud and C. Stump: Vertex barycenter of generalized associahedra, Proc. Amer. Math. Soc. 143 (2015), 2623–2636.
A. Postnikov: Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN 6 (2009), 1026–1106.
A. Postnikov, V. Reiner and L. K. Williams: Faces of generalized permutohedra, Doc. Math. 13 (2008), 207–273.
L. Pournin: The diameter of associahedra, Adv. Math. 259 (2014), 13–42.
N. Reading: Lattice congruences of the weak order, Order 21 (2004), 315–344.
N. Reading: Cambrian lattices, Adv. Math. 205 (2006), 313–353.
N. Reading and D. E. Speyer: Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), 407–447.
S. Shnider and S. Sternberg: Quantum groups: From coalgebras to Drinfeld algebras, Series in Mathematical Physics. International Press, Cambridge, MA, 1993.
D. D. Sleator, R. E. Tarjan and W. P. Thurston: Rotation distance, triangulations, and hyperbolic space, J. Amer. Math. Soc. 1 (1988), 647–681.
L. Solomon: A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255–264.
R. P. Stanley: Hipparchus, Plutarch, Schröder, and Hough, Amer. Math. Monthly 104 (1997), 344–350.
R. P. Stanley: Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
J. Stasheff: Homotopy associativity of H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312.
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.
S. Stella: Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), 121–158.
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.
X. Viennot: Catalan tableaux and the asymmetric exclusion process, in: 19th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2007). 2007.
A. Zelevinsky: Nested complexes and their polyhedral realizations, Pure Appl. Math. Q. 2 (2006), 655–671.
Vincent Pilaud was partially supported by grant MTM2011-22792 of the Spanish MICINN, by the French ANR grant EGOS (12 JS02 002 01), and by European Research Project ExploreMaps (ERC StG 208471).
Carsten Lange was supported by the TEOMATRO grant ANR-10-BLAN 0207.
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Lange, C., Pilaud, V. Associahedra Via Spines. Combinatorica 38, 443–486 (2018). https://doi.org/10.1007/s00493-015-3248-y
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