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The Pólya enumeration theorem can be used to calculate the number of graphs up to isomorphism with a fixed number of vertices, or the generating function of these graphs according to the number of edges they have. For the latter purpose, we can say that a black or present edge has weight 1, while an absent or white edge has weight 0.
According to the De Finetti's theorem, there must be a unique prior distribution such that the joint distribution of observing the sequence is a Bayesian mixture of the Bernoulli probabilities. It can be shown that this prior distribution is a beta distribution with parameters β ( ⋅ ; α , γ ) {\displaystyle \beta \left(\cdot ;\,\alpha ...
[4]: 23–24 The specific topics treated bear witness to the special interests of Pólya (Descartes' rule of signs, Pólya's enumeration theorem), Szegö (polynomials, trigonometric polynomials, and his own work in orthogonal polynomials) and sometimes both (the zeros of polynomials and analytic functions, complex analysis in general).
Pólya’s theorem can be used to construct an example of two random variables whose characteristic functions coincide over a finite interval but are different elsewhere. Pólya’s theorem. If is a real-valued, even, continuous function which satisfies the conditions =,
The complete list of all free trees on 2, 3, and 4 labeled vertices: = tree with 2 vertices, = trees with 3 vertices, and = trees with 4 vertices.. In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the ...
Paul D. Seymour FRS (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory.He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ ...
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Marko Riedel, Pólya's enumeration theorem and the symbolic method; Marko Riedel, Cycle indices of the set / multiset operator and the exponential formula; Harald Fripertinger (1997). "Cycle indices of linear, affine and projective groups". Linear Algebra and Its Applications. 263: 133– 156. doi: 10.1016/S0024-3795(96)00530-7. Harald ...