Descent-Inversion Statistics in Riffle Shuffles

TL;DR

This paper analyzes the statistical properties of riffle shuffles, establishing asymptotic normality for descents and inversions, and exploring longest alternating subsequences, with implications for understanding shuffle randomness.

Contribution

It introduces a novel connection between riffle shuffle statistics and random word statistics, providing asymptotic normality results with explicit convergence rates.

Findings

Asymptotic normality of descents and inversions in riffle shuffles

Convergence rates of order 1/√n in Kolmogorov distance

Results on longest alternating subsequences in riffle-shuffled permutations

Abstract

This paper studies statistics of riffle shuffles by relating them to random word statistics with the use of inverse shuffles. Asymptotic normality of the number of descents and inversions in riffle shuffles with convergence rates of order $1/\sqrt{n}$ in the Kolmogorov distance are proven. Results are also given about the lengths of the longest alternating subsequences of random permutations resulting from riffle shuffles. A sketch of how the theory of multisets can be useful for statistics of a variation of top $m$ to random shuffles is presented.