TAILIEUCHUNG - Báo cáo toán học: "Chromatic statistics for triangulations and Fuß–Catalan complexes"

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 .

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