Probability that a random permutation has no fixed point among the first $k$ elements

Solution 1:

A rigorous proof can be made using the central limit theorem (CLT), as follows.

Since $$ \frac{1}{{n!}}\int_0^1 {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} \to 0 $$ as $n \to \infty$, it suffices to consider the integral from $1$ to $\infty$. Let $c>0$ be arbitrary but fixed, and let $X_i$ be as in my previous answer. Define $f_n {(x)} = x^n e^{-x} / \Gamma(n+1)$, $x>0$, and put $n'=n+1$. Then, since $f_n$ is the density of $\sum\nolimits_{i = 1}^{n'} {X_i }$, $$ \frac{1}{{n!}}\int_1^{n' - c\sqrt {n'} } {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} \leq \int_0^{n' - c\sqrt {n'} } {f_n (x)\,{\rm d}x} = {\rm P}\bigg(\frac{{\sum\nolimits_{i = 1}^{n'} {X_i } - n'}}{{\sqrt {n'} }} \le - c \bigg). $$ By the CLT, since the $X_i$ have unit mean and unit variance, the probability on the right-hand side tends to $\Phi(-c)$ as $n \to \infty$, where $\Phi$ denotes the distribution function of the standard normal distribution. Similarly, $$ \frac{1}{{n!}}\int_{n' + c\sqrt {n'} }^\infty {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} \leq \int_{{n'} + c\sqrt {n'} }^\infty {f_n (x)\,{\rm d}x} = {\rm P}\bigg(\frac{{\sum\nolimits_{i = 1}^{n'} {X_i } - n'}}{{\sqrt {n'} }} > c \bigg), $$ and, by the CLT, the probability on the right-hand side tends to $1 - \Phi(c)$ as $n \to \infty$. Next, note that $$ \bigg(1 - \frac{1}{{n' - c\sqrt {n'} }}\bigg)^k \int_{n' - c\sqrt {n'} }^{n' + c\sqrt {n'} } {f_n (x)\,{\rm d}x} \leq \frac{1}{{n!}}\int_{n' - c\sqrt {n'} }^{n' + c\sqrt {n'} } {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} $$ and $$ \frac{1}{{n!}}\int_{n' - c\sqrt {n'} }^{n' + c\sqrt {n'} } {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} \leq \bigg(1 - \frac{1}{{n' + c\sqrt {n'} }}\bigg)^k \int_{n' - c\sqrt {n'} }^{n' + c\sqrt {n'} } {f_n (x)\,{\rm d}x}. $$ By the CLT, $$ \int_{n' - c\sqrt {n'} }^{n' + c\sqrt {n'} } {f_n (x)\,{\rm d}x} = {\rm P}\bigg(-c \leq \frac{{\sum\nolimits_{i = 1}^{n'} {X_i } - n'}}{{\sqrt {n'} }} \leq c \bigg) \to 2 \Phi(c) -1 $$ as $n \to \infty$. Combining it all, and noting that $$ \bigg(1 - \frac{1}{{n' \pm c\sqrt {n'} }}\bigg)^k = \bigg(1 - \frac{1}{{n' \pm c\sqrt {n'} }}\bigg)^{n(k/n)} \approx e^{ - k/n}, $$ the result follows by choosing $c$ sufficiently large. [Note that first we choose $c$ sufficiently large, then we let $n \to \infty$.]

Solution 2:

Here's a heuristic idea, based on the strong law of large numbers (SLLN). A rigorous proof can be made using the central limit theorem (CLT), as in my new answer.

First write $$ I = \frac{1}{{n!}}\int_0^\infty {x^{n - k} (x - 1)^k e^{ - x} \,{\rm d}x} = \int_0^\infty {\bigg(1 - \frac{1}{x}} \bigg)^k \frac{{x^n e^{ - x} }}{{\Gamma (n + 1)}}\,{\rm d}x. $$ Now, if $X_1,\ldots,X_{n+1}$ are i.i.d. exponential(1) random variables, then their sum $X_1 + \cdots + X_{n+1}$ has gamma density $x^n e^{-x} / \Gamma(n+1)$, $x > 0$. Hence, $$ I = {\rm E} \bigg[ 1 - \frac{1}{{X_1 + \cdots + X_{n + 1} }}\bigg]^k = {\rm E}\bigg[1 - \frac{{n + 1}}{{X_1 + \cdots + X_{n + 1} }}\frac{1}{{n + 1}}\bigg]^k . $$ By SLLN, $(n + 1)^{ - 1} \sum\nolimits_{i = 1}^{n + 1} {X_i }$ converges almost surely to ${\rm E}[X_1 ] = 1$ as $n \to \infty$. So this suggests the approximation $$ I \approx \bigg(1 - \frac{1}{{n + 1}}\bigg)^k = \bigg(1 - \frac{1}{{n + 1}}\bigg)^{n(k/n)} \approx e^{ - k/n}. $$

Solution 3:

Update: This argument only holds for some cases. See italicized additions below.

Let $S_{n,k}$ denote the number of permutations in which the first $k$ elements are not fixed. I published an expository paper on these numbers earlier this year. See "Deranged Exams," (College Mathematics Journal, 41 (3): 197-202, 2010). Aravind's formula is in the paper, as are several others involving $S_{n,k}$ and related numbers.

Theorem 7 (which I also mention in this recent math.SE question) is relevant to this question. It's $$S_{n+k,k} = \sum_{j=0}^n \binom{n}{j} D_{k+j},$$ where $D_n$ is the number of derangements on $n$ elements. See the paper for a simple combinatorial proof of this.

Since $D_n$ grows as $n!$ via $D_n = \frac{n!}{e} + O(1)$ (see Wikipedia's page on the derangement numbers), and if $k$ is much larger than $n$, the dominant terms in the probability $\frac{S_{n+k,k}}{(n+k)!}$ are the $j = n$ and $j = n-1$ terms from the Theorem 7 expression. Thus we have $$\frac{S_{n+k,k}}{(n+k)!} \approx \frac{D_{n+k} + n D_{n+k-1}}{(n+k)!} \approx \frac{1}{e}\left(1 + \frac{n}{n+k}\right) \approx e^{-1} e^{\frac{n}{n+k}} = e^\frac{-k}{n+k},$$ where the second-to-last step uses the first two terms in the Maclaurin series expansion for $e^x$.

Again, this argument holds only for (in my notation) $k$ much larger than $n$.