Adin, Ron M.; Firer, Marcelo; Roichman, Yuval Triangle-free triangulations. (English) Zbl 1278.05245 Adv. Appl. Math. 45, No. 1, 77-95 (2010). Summary: The flip operation on colored inner-triangle-free triangulations of a convex polygon is studied. It is shown that the affine Weyl group \(\tilde C _n\) acts transitively on these triangulations by colored flips, and that the resulting colored flip graph is closely related to a lower interval in the weak order on \(\tilde C _n\). Lattice properties of this order are then applied to compute the diameter. Cited in 4 Documents MSC: 05E18 Group actions on combinatorial structures 06B05 Structure theory of lattices 20F55 Reflection and Coxeter groups (group-theoretic aspects) 52B70 Polyhedral manifolds Keywords:triangulations; flips; group actions; Schreier graphs; Coxeter groups; weak order; Hasse diagrams PDFBibTeX XMLCite \textit{R. M. Adin} et al., Adv. Appl. Math. 45, No. 1, 77--95 (2010; Zbl 1278.05245) Full Text: DOI arXiv References: [1] Björner, A.; Brenti, F., Combinatorics of Coxeter Groups, Grad. Texts in Math., vol. 231 (2005), Springer: Springer New York · Zbl 1110.05001 [2] Conway, J. H.; Coxeter, H. S.M., Triangulated polygons and frieze patterns, Math. Gaz., 57, 87-94 (1973), 175-186 · Zbl 0285.05028 [3] Dehornoy, P., Dual Presentation of Thompson’s Group \(F\) and Flip Distance Between Triangulations, Lect. Notes (June 2008), CIRM [4] Dehornoy, P., On the rotation distance between binary trees (2009), preprint [5] Humphreys, J. E., Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., vol. 29 (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0725.20028 [6] Sagan, B. E., Proper partitions of a polygon and \(k\)-Catalan numbers, Ars Combin., 88, 109-124 (2008) · Zbl 1224.05017 [7] Sleator, D. D.; Tarjan, R. E.; Thurston, W. P., Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc., 1, 647-681 (1988) · Zbl 0653.51017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.