Perfect shuffling with fewer lazy transpositions

TL;DR

This paper investigates the minimal length of sequences of lazy transpositions needed to produce a uniform permutation distribution, providing a construction shorter than the known maximum length, thus advancing understanding of permutation shuffling.

Contribution

The authors present a novel construction of lazy transposition sequences with length approximately two-thirds of the maximum, improving bounds on efficient permutation shuffling.

Findings

Constructed sequences of length ~2/3 * binomial(n,2)

Answered a question about minimal sequence length for uniform distribution

Provided insights into related permutation shuffling problems

Abstract

A lazy transposition $(a,b,p)$ is the random permutation that equals the identity with probability $1-p$ and the transposition $(a,b)\in S_n$ with probability $p$. How long must a sequence of independent lazy transpositions be if their composition is uniformly distributed? It is known that there are sequences of length $\binom{n}2$, but are there shorter sequences? This was raised by Fitzsimons in 2011, and independently by Angel and Holroyd in 2018. We answer this question negatively by giving a construction of length $\frac23 \binom{n}2+O(n\log n)$, and consider some related questions.