Previously we showed that the distribution of fixed points of a random permutation of Image may be NSFW.
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elements behaves asymptotically (in the limit as Image may be NSFW.
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) like a Poisson random variable with parameter Image may be NSFW.
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. As it turns out, this generalizes to the following.
Theorem: As Image may be NSFW.
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of a random permutation of Image may be NSFW.
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.
This is a fairly strong statement which essentially settles the asymptotic description of short cycles in random permutations.
Proof
We will prove pointwise convergence of moment generating functions. First, the Poisson random variable Image may be NSFW.
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with parameter Image may be NSFW.
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is the random variable which takes on non-negative integer values with probabilities
Image may be NSFW.
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.
Image may be NSFW.
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Image may be NSFW.
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which is the exponential generating function of the Touchard polynomials
Image may be NSFW.
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where Image may be NSFW.
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are the Stirling numbers of the second kind. These specialize to the Bell numbers when Image may be NSFW.
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Because we are discussing Image may be NSFW.
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random variables, we should compute a joint moment generating function. The joint moment generating function of Image may be NSFW.
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is
Image may be NSFW.
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.
Back to permutations. By the exponential formula, letting Image may be NSFW.
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denote the number of cycles of length Image may be NSFW.
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, we have
Image may be NSFW.
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which simplifies to
Image may be NSFW.
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.
It is a general and straightforward observation that if Image may be NSFW.
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is a power series with radius of convergence greater than Image may be NSFW.
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, then Image may be NSFW.
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has a power series expansion whose coefficients asymptotically approach Image may be NSFW.
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. Substituting Image may be NSFW.
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, we conclude that the coefficients of
Image may be NSFW.
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asymptotically approach
Image may be NSFW.
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.
But the former is precisely the joint moment generating function of Image may be NSFW.
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, and the latter is precisely the joint moment generating function of independent Poisson random variables with parameters Image may be NSFW.
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. The conclusion follows.
Mean and variance
An exact result which can be deduced using the above methods is that the expected number of Image may be NSFW.
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if Image may be NSFW.
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and Image may be NSFW.
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otherwise. It follows that the total expected number of cycles is the harmonic number
Image may be NSFW.
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.
Since Poisson random variables have the same mean and variance, and by the asymptotic independence statement above, we expect the variance of the total number of cycles to also be asymptotic to Image may be NSFW.
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. This is in fact true, and can be shown using the exponential formula as above.
In The Anatomy of Integers and Permutations, Granville suggested that the decomposition of a random permutation into cycles should be thought of as analogous to the decomposition of a random integer into prime factors. In light of this analogy, the above computation should be thought of as roughly analogous to the Hardy-Ramanujan theorem.