In mathematics, symbolic combinatorics is one of the many techniques of counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. This particular theory is due to Philippe Flajolet and Robert Sedgewick.
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equiva...
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In mathematics, symbolic combinatorics is one of the many techniques of counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. This particular theory is due to Philippe Flajolet and Robert Sedgewick.
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration is the sum of the weights of the objects in the slots. We will first explain how to solve this problem in the labelled and the unlabelled case and use the solution to motivate the creation of classes of combinatorial structures.
The Pólya enumeration theorem solves this problem in the unlabelled case. Let f(z) be the ordinary...
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