The Living Thing / Notebooks : Group theory I would like to learn

\begin{equation*} \sum_{k=1}^n k C_k = n. \end{equation*}

We let \(C := \sum_{k=1}^n C_k\) be the number of cycles (of arbitrary length); this is another natural number-valued random variable, of size at most \(n\)

for fixed \(1 \leq k_1 < \ldots < k_r,\, C_{k_1},\ldots,C_{k_r}\) converge in joint distribution to \(r\) independent Poisson distributions of intensity \(\frac{1}{k_1},\ldots,\frac{1}{k_r}\) respectively…