Random decompositions of Eulerian statistics

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 a rate of convergence of order $n^{-1/2}$ and $n^{-1/3}$ respectively.Comment: 28 page

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence