The results reported by T. Austin, R. Fagen, T. Lehrer, and W. Penney  are extended to include a general recurrence relation for the factorial moments of the distribution. This recurrence relation is solved for the mean and second factorial moments, and it is shown that the method applied may also be used to obtain a general solution for any desired factorial moment of higher order.
"Note on the Factorial Moments of the Distribution of Locally Maximal Elements in a Random Sample." Ann. Math. Statist. 30 (2) 586 - 590, June, 1959. https://doi.org/10.1214/aoms/1177706273