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The combinatorics of spaghetti hoops

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Starting with n cooked spaghetti strands, tie randomly chosen ends together to produce a collection of spaghetti hoops. What is the expected number of hoops? What can be said about the distribution of the number of hoops of length 1, 2, …? What is the behaviour of the longest hoops when n is large? What is the probability that all the hoops have different lengths? Questions like this appear in many guises in many areas of mathematics, the connection being their relation to the Ewens Sampling Formula (ESF). I will describe a number of related examples, including prime factorisation, random mappings and random permutations, illustrating the central role played by the ESF . I will also discuss methods for simulating decomposable combinatorial structures by exploiting another wonder of the ESF world, namely the Feller Coupling. Analysis of a children’s playground game shows that apparently small departures from the Feller model can open up a number of unsolved problems.

This talk is part of the Cambridge Philosophical Society series.

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