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An Elementary Proof of the Cayley Formula Using Random Maps

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arxiv 1409.1614 v1 pith:6MV35522 submitted 2014-09-04 math.CO

An Elementary Proof of the Cayley Formula Using Random Maps

classification math.CO
keywords formulacayleyelementelementaryproofrandombijectivecalculation
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Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an elementary calculation of the probability that a cycle occurs in a random map from an $n$-element set to an $n+1$-element set.

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