  For $N\in\mathbb{N}$, let $\pi_N$ be the law of the number of fixed points of
a random permutation of $\{1, 2, ..., N\}$. Let $\mathcal{P}$ be a Poisson law
of parameter 1.A classical result shows that $\pi_N$ converges to $\mathcal{P}$
for large $N$ and indeed in total variation $$\left\Vert
\pi_N-\mathcal{P}\right\Vert_{\mathrm{tv}} \leq \frac{2^N}{(N+1)!}$$ This
implies that $\pi_N$ and $\mathcal{P}$ can be coupled to at least this
accuracy. This paper constructs such a coupling (a long open problem) using the
machinery of intertwining of two Markov chains. This method shows promise for
related problems of random matrix theory.
