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On a constant arising in the asymtotic theory of symmetric groups
Zv. Ignatov Bulgaria
Abstract:
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.
Received: 14.01.1980