On a constant arising in the asymtotic theory of symmetric groups

Let $x_1(g)\ge x_1(g)\ge\dots$ be the lengths of the cycles of the permutation $g\in S_n$ and
$$\widetilde\Sigma=\{(\sigma_1,\sigma_2,\dots):\,\sigma_1\ge\sigma_2\ge\dots,\ \sigma_1+\sigma_2+\dots=1\}$$
The uniform probability distribution on $S_n$ and the map
$$S_n\to\widetilde\Sigma:\,g\to(n^{-1}x_1(g),\,n^{-1}x_2(g),\dots)$$
generate a probability distribution on $\widetilde\Sigma$. We investigate some properties of this distribution when $n\to\infty$. In particular, we prove that the constant introduced in [1], [2] coincides with the Euler constant.