On the statistics of random-to-top shuffles

TL;DR

This paper establishes limit theorems for key permutation statistics in random-to-top shuffles, providing new combinatorial proofs and solving open problems in the field.

Contribution

It introduces analytic methods and combinatorial decompositions to analyze fixed points, descents, and inversions in iterated random-to-top shuffles, addressing open questions.

Findings

Proves limit theorems for fixed points, descents, and inversions.

Provides new combinatorial proofs for expected fixed points and inversions.

Solves an open problem of Pehlivan and answers a question of Diaconis and Fulman.

Abstract

We prove limit theorems for the number of fixed points, descents, and inversions of iterated random-to-top shuffles in two asymptotic regimes. Our proofs are analytic, and they utilize new combinatorial decompositions that represent each statistic as a randomly indexed statistic of a uniformly random permutation. This perspective gives new combinatorial proofs of the expected number of fixed points and inversions. In particular, we solve an open problem of Pehlivan on fixed points, and we answer a question of Diaconis and Fulman on inversions.