Perfect shuffling by lazy swaps

TL;DR

This paper characterizes the shortest sequences of lazy swaps that produce a uniform random permutation, providing a unique probability assignment for each reduced word of the reverse permutation.

Contribution

It offers a precise characterization of minimal lazy transposition sequences for uniform permutation generation and addresses the uniqueness of probability assignments.

Findings

Unique probability assignment for each reduced word of the reverse permutation.

Characterization of minimum-length lazy transposition sequences.

Open problem on minimum length without simplicity condition.

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.