Sum of positions of records in random permutations: asymptotic analysis

This statistic, i.e. the sum of positions of records, has been the object of recent interest in the literature. Using the saddle point method, we obtain from the generating function of the sum of positions of records in random permutations and Cauchy’s integral formula, asymptotic results in central and non-central regions. In the non-central region, we derive asymptotic expansions generalizing some results by Kortchemski. In the central region, we obtain a limiting distribution related to Dickman’s function. This paper fits within the framework of Analytic Combinatorics.