Reconfiguration and Enumeration of Optimal Cyclic Ladder Lotteries

TL;DR

This paper studies how to reconfigure and count optimal cyclic ladder lotteries, focusing on permutations and displacement vectors, to understand their structure and possible transformations.

Contribution

It introduces methods for reconfiguration and enumeration of optimal cyclic ladder lotteries based on permutations and displacement vectors.

Findings

Developed algorithms for reconfiguration of optimal lotteries.

Provided enumeration techniques for possible lottery configurations.

Analyzed the structural properties of cyclic ladder lotteries.

Abstract

A ladder lottery, known as ``Amidakuji'' in Japan, is a common way to decide an assignment at random. In this paper, we investigate reconfiguration and enumeration problems of cyclic ladder lotteries. First, when a permutation $\pi$ and an optimal displacement vector $\mathbf{x}$ are given, we investigate the reconfiguration and enumeration problems of the ``optimal'' cyclic ladder lotteries of $\pi$ and $\mathbf{x}$. Next, for a give permutation $\pi$ we consider reconfiguration and enumeration problems of the optimal displacement vectors of $\pi$.