On the Golomb-Dickman constant under Ewens sampling

Abstract: We define a generalized Golomb-Dickman constant $λθ$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $θ> 0$. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution, we obtain an explicit integral representation for $λθ$ in terms of the exponential integral. The dependence of $λθ$ on $θ$ reflects the transition between regimes dominated by long cycles (small $θ$) and those with many small cycles (large $θ$). Our result can be viewed as an extension of the classical calculations of Shepp and Lloyd to the Ewens setting by relatively elementary means. A figure and a table of numerical values of $λθ$ are included.