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It is general consensus that Combinatorics has developed into a full-fledged mathematical discipline whose beginnings as a charming pastime have long since been left behind and whose great signifi cance for other branches of both pure and applied mathematics is only beginning to be realized. The last ten years have witnessed a tremendous outburst of activity both in relatively new fields such as Coding Theory and the Theory of Matroids as well as in' more time honored endeavors such as Generating Functions and the Inver sion Calculus. Although the number of text books on these subjects is slowly increasing, there is also a great need for up-to-date surveys of the main lines of research designed to aid the beginner and serve as a reference for the expert. It was the aim of the Advanced Study Institute "Higher Combinatorics" in Berlin, 1976, to help fulfill this need. There were five sections: I. Counting Theory, II. Combinatorial Set Theory and Order Theory, III. Matroids, IV. Designs and V. Groups and Coding Theory, with three principal lecturers in each section. Expanded versions of most lectures form the contents of this book. The Institute was designed to offer, especially to young researchers, a comprehen sive picture of the most interesting developments currently under way. It is hoped that these proceedings will serve the same purpose for a wider audience.
List of contents
I - Counting Theory.- Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials.- Distributions Eulériennes et Mahoniennes sur le Groupe des Permutations.- Cohen-Macaulay Complexes.- II - Combinatorial Set Theory and Order Theory.- Acyclic Orientations.- Problems in Combinatorial Set Theory.- Aspects of Finite Lattices.- III - Matroids.- Geometries Associated with Partially Ordered Sets.- Transversal Matroids and Related Structures.- Matroids as the Study of Geometrical Configurations.- IV - Designs.- Partitions of X2 and Other Topics.- Combinatorics of Partial Geometries and Generalized Quadrangles.- Edge Decompositions of Colored Graphs.- V - Groups and Coding Theory.- Planes and Biplanes.- Ovals from the Point of View of Coding Theory.- Permutation Groups on Unordered Sets.- Codes and Designs.
