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Coin Turning Games on Partially Ordered Sets

Masao Ishikawa, Toyokazu Ohmoto, Hiroyuki Tagawa, Yoshiki Takayama

Abstract

A finite impartial game is a two-player game in which the players take turns making moves and the game ends after finitely many moves. In this paper, we study a class of finite impartial games introduced by H.~Lenstra, which we call coin turning games. We focus on two typical classes of coin turning games, namely the order ideal games and the rulers, distinguished by their choices of turning sets. For several posets arising from enumerative combinatorics, we determine the Sprague-Grundy functions. In particular, we determine the Sprague-Grundy function of the order ideal game on the ASM poset, introduced by J.~Striker in connection with the alternating sign matrices.

Coin Turning Games on Partially Ordered Sets

Abstract

A finite impartial game is a two-player game in which the players take turns making moves and the game ends after finitely many moves. In this paper, we study a class of finite impartial games introduced by H.~Lenstra, which we call coin turning games. We focus on two typical classes of coin turning games, namely the order ideal games and the rulers, distinguished by their choices of turning sets. For several posets arising from enumerative combinatorics, we determine the Sprague-Grundy functions. In particular, we determine the Sprague-Grundy function of the order ideal game on the ASM poset, introduced by J.~Striker in connection with the alternating sign matrices.
Paper Structure (25 sections, 29 theorems, 42 equations, 9 figures, 4 tables)

This paper contains 25 sections, 29 theorems, 42 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Finite Impartial Games
  3. Basic notions of finite impartial games
  4. Combined game
  5. NIM multiplication
  6. Partially Ordered Sets
  7. Preliminaries on Posets
  8. The Poset of Set Partitions
  9. A Poset Associated with Alternating Sign Matrices
  10. Sprague-Grundy Theory of Coin Turning Games
  11. Coin Turning Games
  12. The Fundamental Theorem of Coin Turning Games
  13. The Product Theorem for Coin Turning Games
  14. Two Typical Classes of Coin Turning Games
  15. Order Ideal Games
  16. ...and 10 more sections

Key Result

Theorem 1.1

For each $(x, y, z) \in \boldsymbol{A}_{n}$, the Sprague--Grundy value of the order ideal game on $\boldsymbol{A}_{n}$ is given by

Figures (9)

  • Figure 1: $B_3(2)$
  • Figure 2: $\Pi_{4}$
  • Figure 3: $\boldsymbol{A}_{5}$
  • Figure 4: Poset $D_{12}$ of divisors of $12$
  • Figure 5: The Grundy values of $B_3(2)$
  • ...and 4 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1: WWONAGLenstraSatoSiegel
  • Definition 2.2: Sprague-Grundy function GrundySprague
  • Definition 2.3
  • Definition 2.4: ONAGLenstraSato
  • Theorem 2.1: Lenstra
  • Definition 2.5: ONAGLenstraSato
  • Definition 3.1: ec1
  • Proposition 3.1: St
  • Definition 4.1: Coin Turning Game LenstraSato
  • ...and 28 more