Coin Turning Games on Partially Ordered Sets

TL;DR

This paper investigates coin turning games on partially ordered sets, analyzing their Sprague-Grundy functions for specific combinatorial posets, including those related to alternating sign matrices, to understand their game-theoretic properties.

Contribution

It introduces and studies coin turning games on posets, providing explicit Sprague-Grundy functions for several classes, notably the ASM poset, advancing combinatorial game theory.

Findings

Determined Sprague-Grundy functions for order ideal games on various posets.

Established the Sprague-Grundy function for the ASM poset.

Analyzed the structure of coin turning games in combinatorial contexts.

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.