Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The centers of gravity of the associahedron and of the permutahedron are the same. | The centers of gravity of the associahedron and of the permutahedron are the same Christophe Hohlweg Jonathan Lortie LaCIM et Departement de Mathematiques LaCIM et Departement de Mathematiques Universite du Quebec a Montreal Universite du Quebec a Montreal CP 8888 Succ. Centre-Ville CP 8888 Succ. Centre-Ville Montreal Quebec H3C 3P8 CANADA Montreal Quebec H3C 3P8 CANADA onathan@ Annie Raymond Berlin Mathematical School Strasse des 17. Juni 136 Berlin 10623 Germany raymond@ Submitted Mar 8 2010 Accepted May 4 2010 Published May 14 2010 Mathematics Subject Classification 05E18 05E99 Abstract In this article we show that Loday s realization of the associahedron has the the same center of gravity as the permutahedron. This proves an observation made by F. Chapoton. We also prove that this result holds for the associahedron and the cyclohedron as realized by the first author and C. Lange. 1 Introduction. This article is the continuation of previous work 7 8 1 devoted to the study of generalized associahedra via geometric and combinatorial tools arising from finite Coxeter groups. In 1963 J. Stasheff discovered a cell complex 21 22 of great importance in algebraic topology geometric topology and combinatorics 2 6 10 . This cell complex can be realized as a simple n 1-dimensional convex polytope in Rn the associahedron. Many realizations of the associahedron were given over the last thirty years 13 . In This work is supported by FQRNT and NSERC. It is the result of a summer undergraduate research internship supported by LaCIM THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R72 1 2004 J. L. Loday 9 computed the classical realization of the associahedron given by S. Shnider and S. Sternberg in 19 . Loday gave a beautiful combinatorial algorithm to compute the integer coordinates of the vertices of the associahedron and showed that his realization can be obtained naturally from the classical permutahedron .
