Random Decompositions of Eulerian Statistics

Abstract

This paper develops methods to study the distribution of Eulerian statistics defined by second-order recurrence relations. We define a random process to decompose the statistics over compositions of integers. It is shown that the numbers of descents in random involutions and in random derangements are asymptotically normal with rates of convergence $\mathcal{O} (n^{-1/2})$ and $\mathcal{O}(n^{-1/3})$ respectively.