On the Golomb-Dickman constant under Ewens sampling

We define a generalized Golomb–Dickman constant ${\lambda }_{\theta }$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $\theta >0$. Exploiting the independence properties of Kingman’s Poisson process construction of the Poisson–Dirichlet distribution, we obtain an explicit integral representation for ${\lambda }_{\theta }$ in terms of the exponential integral. The dependence of ${\lambda }_{\theta }$ on $\theta$ reflects the transition between regimes dominated by long cycles (small $\theta$) and those with many small cycles (large $\theta$). We also derive the asymptotic behavior of ${\lambda }_{\theta }$ for small and large $\theta$ and illustrate our results with numerical computations, Monte Carlo simulations of the Hoppe urn, and an application.