On the Golomb–Dickman constant under Ewens sampling

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). We also derive the asymptotic behavior of λθ for small and large θ and illustrate our results with numerical computations, Monte Carlo simulations of the Hoppe urn, and an application.