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Tag Archives: generating functions
Burnside’s Lemma and Polya Enumeration Theorem (2)
[ Acknowledgement: all the tedious algebraic expansions in this article were performed by wolframalpha. ] Counting Graphs One of the most surprising applications of Burnside’s lemma and Polya enumeration theorem is in counting the number of graphs up to isomorphism. … Continue reading →
Posted in Notes
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Tagged advanced, burnside's lemma, combinatorics, generating functions, group actions, group theory, polya enumeration theorem, stirling numbers, symmetries
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Random Walk and Differential Equations (I)
Consider discrete points on the real line, indexed by the integers … -3, -2, -1, 0, 1, 2, … . A drunken man starts at position 0 and time 0. At each time step, he may move to the left … Continue reading →
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Tagged generating functions, heat equation, partial differential equations, random walk
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Power Series and Generating Functions (IV) – Exponential Generating Functions
Note: this article is noticeably more difficult than the previous instalments. The reader is advised to be completely comfortable with generating functions before proceeding. We’ve already seen how generating functions can be used to solve some combinatorial problems. The nice … Continue reading →
Posted in Notes
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Tagged advanced, combinatorics, exponential generating functions, generating functions, intermediate, notes
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Power Series and Generating Functions (III) – Partitions
One particularly fruitful application of generating functions is in partition numbers. Let n be a positive integer. A partition of n is an expression of n as a sum of positive integers, where two expressions are identical if they can be obtained from each … Continue reading →
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Tagged combinatorics, generating functions, intermediate, notes, partition number, partitions, power series
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Power Series and Generating Functions (I): Basics
[ Background required: basic combinatorics, including combinations and permutations. Thus, you should know the formulae and and what they mean. Also, some examples / problems may require calculus. ] Note: this post is still highly relevant to competition-mathematics. 🙂 To … Continue reading →
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Tagged basic, combinatorics, generating functions, intermediate, polynomials, power series
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Mathematics and Such 