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: Chromatic statistics for triangulations and Fuß–Catalan complexes. | Chromatic statistics for triangulations and Fuh-Catalan complexes R. Bacher Universite Grenoble I CNRS UMR 5582 Institut Fourier 100 rue de maths BP 74 F-38402 St. Martin d Heres Cedex France http bacher C. Krattenthaler Fakultat fur Mathematik Universitat Wien Nordbergstrafie 15 A-1090 Vienna Austria http kratt Submitted Jan 13 2011 Accepted Jul 12 2011 Published Jul 22 2011 2010 Mathematics Subject Classification Primary 05A15 Secondary 05A19 Abstract We introduce Fufi-Catalan complexes as d-dimensional generalisations of triangulations of a convex polygon. These complexes are used to refine Catalan numbers and Fufi-Catalan numbers by introducing colour statistics for triangulations and Fufi-Catalan complexes. Our refinements consist in showing that the number of triangulations respectively of Fufi-Catalan complexes with a given colour distribution of its vertices is given by closed product formulae. The crucial ingredient in the proof is the Lagrange-Good inversion formula. Keywords Catalan number Fufi-Catalan number triangulation Fufi-Catalan complex barycentric subdivision Schlegel diagram vertex colouring simplicial complex Lagrange-Good inversion formula. Research partially supported by the Austrian Science Foundation FWF grants Z130-N13 and S9607-N13 the latter in the framework of the National Research Network Analytic Combinatorics and Probabilistic Number Theory. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P152 1 1 Introduction Catalan and FuB Catalan numbers The sequence Cn n 0 of Catalan numbers 1 1 2 5 14 42 132 429 1430 4862 16796 58786 . see 13 sequence A108 defined by C 1 2n 1 2n n n 1 nJ n n 1 is ubiquitous in enumerative combinatorics. Exercise in 14 contains a list of 66 sequences of sets enumerated by Catalan numbers with many more in the addendum 15 . In particular there are n i 2n triangulations of a convex polygon1 with n 2 vertices see 14 Ex. . Even many years before .