Perfect shuffling by lazy swaps

Omer Angel,Alexander E Holroyd

doi:10.1214/18-ecp151

Abstract

We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation. For every reduced word of the reverse permutation there is exactly one valid way to assign probabilities to the transpositions. It is an open problem to determine the minimum length of such a sequence when the simplicity condition is dropped.

Highlights

Let Sn be the symmetric group of all permutations of 1, . . . , n, with composition given by(i) := σ(τ (i)) for 1 ≤ i ≤ n
We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation
A lazy transposition with parameters (a, b, p) is a random permutation T that with probability p equals the transposition t(a, b) := ∈ Sn exchanging the elements in positions a and b, and otherwise equals the identity id ∈ Sn

Summary

Introduction

Let Sn be the symmetric group of all permutations of 1, . . . , n, with composition given by (στ )(i) := σ(τ (i)) for 1 ≤ i ≤ n. Call a sequence of parameters (ai, bi, pi)ki=1 a sweep (of order n and length k) if the composition of independent lazy transpositions of Sn with these parameters, π say, has the property that its last element π(n) is uniformly distributed on 1, . Applying the inductive construction of the previous paragraph to this example gives a simple transposition shuffle which is a special case of Construction 1.2. The construction gives examples of transposition shuffles of length n 2 in which two of the probabilities pi may be simultaneously altered to give another transposition shuffle (compare Theorem 1.3 and Proposition 1.5).

Background

Simple transpositions

General transpositions