Smoothness and decay properties of the limiting Quicksort density function

Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used by Quicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f , and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16. AMS 2000 subject classifications. Primary 68W40; secondary 68P10, 60E05, 60E10.