A central limit theorem for descents and major indices in fixed conjugacy classes of Sn

Abstract

In this paper, we show that the joint distribution of descents and major indices in conjugacy class is asymptotically bivariate normal. This generalizes the authors' previous work on the asymptotical normality of descents in conjugacy classes, where the asymptotic parameters depended only on the density of fixed points. The result is achieved by two key ingredients; one is a variation of the continuity theorem in which the region of pointwise convergence of moment generating functions (m.g.f.s) can be chosen arbitrarily, and the other is a uniform estimate on the m.g.f.s of the descents/major-index pairs in conjugacy classes. As a byproduct, the authors obtain a generating function for the descents and major indices of a given conjugacy class.