Exercise 28, § 5

For every integer $k\geqslant 0$, let $b_k$ denote the number of equivalence relations on the set $\{1, . . . , k\}$.

For every $\lambda >0$, let $\varpi_{\lambda}$ denote the Poisson measure with parameter $\lambda$ on $\mathbf{R}($cf. exercise 41 of II, p. 282), that is, the measure of total mass 1 supported on $\mathbf{N}$ such that $\varpi_{\lambda}(j) =e^{-\lambda}\lambda^j/j$! for every $j\in \mathbf{N}$.

a) The relation

$j=0$

is valid for every integer $k\in \mathbf{N}$.

b) For every function $f:\mathbf{N}\rightarrow \mathbf{C}$ such that the function $x\mapsto xf(x)$ is $\varpi$-integrable, one has

("Chen's formula").

Moreover, if $\mu$ is a bounded measure on $\mathbf{R}$ satisfying this property, then there exists $c\in \mathbf{R}$ such that $\mu=c\varpi_{\lambda}$.

c) For every $k\in \mathbf{N}$, the function $x\mapsto x^k$ is $\varpi_1$-integrable and one has

Moreover, the measure $\varpi_1$ is the unique positive measure on $\mathbf{R}$ satisfying these equations.

d) For every integer $k\geqslant 0$, one has

$b_k=\frac{1}{e}\sum_{j\geqslant 0}jj^n$!

("Dobiński's formula") and

$b_k=2e\pi k$! $\mathscr{I}(\int_0^{\pi}e^{e^{eit}}$ sin($kt$)$dt)$

("Cesàro's formula").

e) For every integer $n\geqslant 1$, let $\mu_n$ denote the positive measure on $\mathbf{R}$ which is the image of the normalized Haar measure of the symmetric group $S_n$ under the mapping $f_n$ from $S_n$ into $\mathbf{R}$ which to $\sigma \in S_n$ associates the number of fixed points of $\sigma$. Let $k\in \mathbf{N}$. For every integer $n\geqslant k$, one has

f) The sequence $(\mu_n)$ converges narrowly as $n\rightarrow +\infty$ to the Poisson measure $\varpi_1$.