On a Markov construction of couplings

For $N\in \mathbb{N}$, let ${\pi }_N$ be the law of the number of fixed points of a random permutation of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},N\}$. 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 $${\parallel {\pi }_N-\mathcal{P}\parallel }_{\mathrm{t}\mathrm{v}}\le \frac{2^N}{(N+1)!}$$ 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.