Random decompositions of Eulerian statistics

TL;DR

This paper introduces a new method for analyzing the distribution of Eulerian statistics using random decompositions, demonstrating asymptotic normality for descents in involutions and derangements.

Contribution

It develops a novel approach to study Eulerian statistics via random decompositions and establishes asymptotic normality results for specific permutation classes.

Findings

Descents in involutions are asymptotically normal with rate O(n^{-1/2})

Descents in derangements are asymptotically normal with rate O(n^{-1/3})

Provides a framework for analyzing Eulerian statistics with second-order recurrence relations

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.