The associahedron and triangulations of the \(n\)-gon. (English) Zbl 0682.52004

For a convex \(n\)-gon \(P_n\), \(n\geq 3\), in the plane let \(\Sigma_n\) denote the collection of all sets of mutually noncrossing diagonals of \(P_n\). The author proves that \(\Sigma_n\) is isomorphic to the boundary complex of some \((n-3)\)-dimensional simplicial convex polytope, and that this polytope can be geometrically realized with the dihedral group \(D_n\) as its symmetry group.


52B99 Polytopes and polyhedra
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[1] Brøndsted, A., An Introduction to Convex Polytopes (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0509.52001
[2] Cayley, A., On the partitions of a polygon, Proc. Lond. Math. Soc, 22, 1, 237-262 (1890-1891)
[3] Dantzig, G. B.; Hoffman, A. J.; Hu, T. C., Triangulations (tilings) and certain block triangular matrices, Math. Programm, 31, 1-14 (1985) · Zbl 0571.90098
[4] Dörrie, H., Euler’s problem of polygon division, (100 Great Problems of Elementary Mathematics (transl. D. Antin) (1965), Dover: Dover New York), 21-27
[5] Gardner, M., (Catalan numbers, Time Travel and Other Mathematical Bewilderments (1988), W. H. Freeman: W. H. Freeman New York), 253-266 · Zbl 0641.00005
[6] Grünbaum, B., Convex Polytopes (1967), Interscience: Interscience London · Zbl 0163.16603
[7] M. Haiman, Constructing the associahedron, manuscript.; M. Haiman, Constructing the associahedron, manuscript.
[8] Huguet, D.; Tamari, D., La structure polyedrale des complexes de parenthésages, J. Comb., Inf. & Syst. Sci., 3, 69-81 (1978) · Zbl 0441.68080
[9] Kirkman, T. P., On the \(k\)-partitions of the \(r\)-gon and \(r\)-ace, Phil. Trans. R. Soc. Lond., 147, 217-272 (1857)
[10] Lee, C. W., (Some notes on triangulating polytopes. Some notes on triangulating polytopes, Proceedings, 3. Kolloquium über Diskrete Geometrie, Institut für Mathematik (May 1985), Universität Salzburg: Universität Salzburg Salzburg, Austria), 173-181 · Zbl 0571.52009
[11] McMullen, P.; Shephard, G. C., Convex Polytopes and the Upper Bound Conjecture (1971), Cambridge University Press · Zbl 0217.46702
[12] Perles, M. A., List of problems, Conference on Convexity, Oberwolfach (July 1984), Federal Republic of Germany
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