Reconfiguration and Enumeration of Optimal Cyclic Ladder Lotteries

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 $\pi$.