A probabilistic approach to the descent statistic

Abstract

We present a probabilistic approach to studying the descent statistic based upon a two-variable probability density. This density is log concave and, in fact, satisfies a higher order concavity condition. From these properties we derive quadratic inequalities for the descent statistic. Using Fourier series, we give exact expressions for the Euler numbers and the alternating r-signed permutations. We also obtain a probabilistic interpretation of the sin function. © 2002 Elsevier Science (USA).