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Reconfiguration and Enumeration of Optimal Cyclic Ladder Lotteries

Yuta Nozaki, Kunihiro Wasa, Katsuhisa Yamanaka

TL;DR

This work extends the theory of ladder lotteries to the cyclic (affine) setting, formalizing optimality as the minimum number of bars and linking structures to the affine symmetric group. It proves that reconfiguration graphs for cyclic ladders with fixed optimal displacement vectors are connected and provides exact shortest reconfiguration lengths via tangled triples, while also delivering polynomial-delay enumeration methods. By introducing and exploiting optimal displacement vectors and max–min contractions, the authors show how to partition the problem and efficiently enumerate all optimal cyclic ladders. The results illuminate the combinatorial structure of cyclic ladder lotteries, enable practical reconfiguration and enumeration algorithms, and connect to pseudoline arrangements on a cylinder and to reverse-search techniques. This has implications for understanding reconfiguration complexity in related Coxeter-group-inspired models and for efficient randomization and counting within cyclic ladder frameworks.

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 $π$ and an optimal displacement vector $\mathbf{x}$ are given, we investigate the reconfiguration and enumeration problems of the ``optimal'' cyclic ladder lotteries of $π$ and $\mathbf{x}$. Next, for a give permutation $π$ we consider reconfiguration and enumeration problems of the optimal displacement vectors of $π$.

Reconfiguration and Enumeration of Optimal Cyclic Ladder Lotteries

TL;DR

This work extends the theory of ladder lotteries to the cyclic (affine) setting, formalizing optimality as the minimum number of bars and linking structures to the affine symmetric group. It proves that reconfiguration graphs for cyclic ladders with fixed optimal displacement vectors are connected and provides exact shortest reconfiguration lengths via tangled triples, while also delivering polynomial-delay enumeration methods. By introducing and exploiting optimal displacement vectors and max–min contractions, the authors show how to partition the problem and efficiently enumerate all optimal cyclic ladders. The results illuminate the combinatorial structure of cyclic ladder lotteries, enable practical reconfiguration and enumeration algorithms, and connect to pseudoline arrangements on a cylinder and to reverse-search techniques. This has implications for understanding reconfiguration complexity in related Coxeter-group-inspired models and for efficient randomization and counting within cyclic ladder frameworks.

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 and an optimal displacement vector are given, we investigate the reconfiguration and enumeration problems of the ``optimal'' cyclic ladder lotteries of and . Next, for a give permutation we consider reconfiguration and enumeration problems of the optimal displacement vectors of .
Paper Structure (10 sections, 21 theorems, 13 equations, 10 figures, 2 algorithms)

This paper contains 10 sections, 21 theorems, 13 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Longest elements
  4. Reconfiguration and enumeration of cyclic ladder lotteries with optimal displacement vectors
  5. Reachability
  6. Shortest reconfiguration
  7. Enumeration
  8. Reconfiguration and enumeration of optimal cyclic ladder lotteries
  9. Reconfiguration
  10. Enumeration

Key Result

Lemma 1

Let $\pi$ be a permutation in $\mathfrak{S}_n$. Then, where $\mathcal{D}(\pi)$ is the set of the optimal displacement vectors of $\pi$.

Figures (10)

  • Figure 1: (a) An optimal ladder lottery of the permutation $(5,1,4,6,2,3)$. (b) A cyclic ladder lottery of the permutation $(4,7,5,3,1,2,6,8)$ and (c) its representation as a pseudoline arrangement. (d) A cyclic ladder lottery obtained from (b) by applying a braid relation to the triple $(2,3,4)$.
  • Figure 2: (a) An optimal cyclic ladder lottery of the permutation $(4,2,6,1,5,3)$ and its optimal displacement vector $(-3,0,-3,3,0,3)$ and (b) an optimal cyclic ladder lottery of the same permutation and its optimal displacement vector $(-3,0,3,-3,0,3)$.
  • Figure 3: Illustrations of (a) a left tangled triple and (b) a right tangled triple.
  • Figure 4: The pseudoline arrangements corresponding to the displacement vectors in Lemma \ref{['lem:longest']} when $m=4$.
  • Figure 5: Illustrations for (a) Case 1 and (b) Case 2.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Lemma 1
  • Proposition 2
  • Lemma 3
  • proof
  • proof : Proposition \ref{['prop:longest']}
  • Remark 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 29 more