$\mathbf{SETUP}$
From this previous question, I quote Cauchy's formula for the number of all possible cycle types
\begin{align} N_{\lambda} = \frac{n!} {1^{\alpha_1} 2^{\alpha_2} ... n^{\alpha_n} \times \alpha_1! \alpha_2! ... \alpha_n!} = \frac{n!}{\Lambda_\lambda} \end{align}
and in this other previous question asked about why the remarkable result exists: that this sum tends to the exponential number raised to the inverse power of the $k$th harmonic number:
\begin{align} \mathcal{E}_k \equiv \lim_{n \rightarrow \infty} \sum_{\lambda \in P_{n,k}} \frac{1}{\Lambda_\lambda} = e^{-H_k} \end{align}
where the sum is over all the partitions of $n$ using integers strictly $> k$, $P_{n,k}$.
$\mathbf{RESULT?}$
Thus we can write the harmonic number as the logarithm of this limit:
\begin{align} H_k = -\ln \mathcal{E}_k \end{align}
which clarifies the connection to the Euler-Mascheroni constant $\gamma = 0.57721...$ from one of its definitions as
\begin{align} \gamma = \lim_{n \rightarrow \infty}(H_n - \ln n) \end{align} which, when written with an error term is:
\begin{align} H_n & = \ln n + \gamma + \epsilon \end{align} with the error being:
\begin{align} \epsilon = \frac{1}{2n} - \frac{1}{12n^2} + \frac{1}{120n^4} - \delta \end{align} with its final bound somewhere $0 < \delta < \frac{1}{252n^6}$, all according to the Wikipedia page.
Thus we can write:
\begin{align} \gamma + \epsilon & = H_n - \ln n \\ \gamma + \epsilon & = -\ln \mathcal{E}_k - \ln n \\ \gamma + \epsilon & = -\ln(\mathcal{E}_k n) \end{align}
$\mathbf{QUESTION}$
Is this result relationship interesting or useful in any way? What explains this deep connection of partitions to Euler-Mascheroni?
The Arratia and Tavaré paper
explains similar links in its section 5.2 (page 1586), but doesn't take it further, nor am I familiar enough with its notation to really understand what it's trying to say.
