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Partition patterns under group action. (English) Zbl 0795.05019

We study enumerative problems concerning partition patterns and types under group action in a general framework \((g\)-graphs and GPO-sets). Hence the results can be applied to graphs, digraphs, hypergraphs, etc.
Our aim is the generalization of classic concepts and properties, both from partitions and Pólya’s theory, to a more detailed structural analysis of a partition. In fact a partition of a graph-like structure (defined as set of edges) can be usually characterized by the types (with respect to a given group) of its blocks, but partitions with the same family of block types can have different families of blocks obtained as unions of the blocks of the partitions (a different “pattern”). The first step is to distinguish partitions with different patterns and to study classic partition theory properties for such partition patterns. The generalization of concepts of Pólya’s theory will be considered only for an application to the problem of pseudo-similarity.

MSC:

05A99 Enumerative combinatorics
05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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