The Ungar Games

TL;DR

This paper introduces the Ungar game played on finite lattices, analyzes winning strategies for specific lattice classes, and provides enumerative results and complexity discussions for these combinatorial structures.

Contribution

It defines the Ungar game on finite lattices, characterizes winning positions in various lattice classes, and derives enumerative formulas for Eeta wins in these structures.

Findings

Number of Eeta wins in weak order on $S_n$ is $O(0.95586^nn!)$

Characterization of Eeta wins in Young's lattice intervals including rectangles and root posets

Enumeration of Eeta wins in Tamari lattices

Abstract

Let $L$ be a finite lattice. An Ungar move sends an element $x\in L$ to the meet of $\{x\}\cup T$, where $T$ is a subset of the set of elements covered by $x$. We introduce the following Ungar game. Starting at the top element of $L$, two players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say $L$ is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on $L$. We first prove that the number of principal order ideals in the weak order on $S_n$ that are Eeta wins is $O(0.95586^nn!)$. We then consider a broad class of intervals in Young's lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-$A$ root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.