Martingales and descent statistics

TL;DR

This paper introduces martingale-based techniques to analyze descent statistics in permutations, extending Berry-Esseen theorem applicability and proving asymptotic normality with explicit error bounds for various related combinatorial statistics.

Contribution

It relaxes a key assumption in the Berry-Esseen theorem to extend its scope to martingales with time-dependent variances, providing new proofs and broad applicability.

Findings

Number of descents in permutations is asymptotically normal with error order 1/√n

Techniques apply to various descent-related statistics satisfying recurrence relations

Extended Berry-Esseen theorem to martingales with time-dependent variances

Abstract

This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.