The Ungar Games

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.

Figures (6)

• Figure 1: Allowable moves in Chomp (left) and in Nibble (right). Starting positions for which the second player has a winning strategy are indicated in gold.
• Figure 2: Five points in the plane are numbered ${\color{red}1},{\color{MyPurple}2},{\color{MyOrange}3},{\color{NormalGreen}4},{\color{SkyBlue}5}$. One can project the points onto a line and read the ordering of the projections along the line to obtain a permutation. When the line rotates, the associated permutation changes via an Ungar move.
• Figure 3: A lattice with $7$ Atniss wins (labeled A) and $5$ Eeta wins (labeled E). The entire lattice is an Atniss win.
• Figure 4: Deleting the (red) steps that lie on the boundary of $\delta_{10}$ or the $x$-axis or $y$-axis breaks a lattice path into $3$ smaller lattice paths.
• Figure 5: An order ideal of the shifted staircase $\mathrm{SS}_5$ is shown in red. This order ideal is uniquely determined by a path of up and down steps lying just above it, and that path corresponds to the length-$5$ binary string ${\color{SkyBlue}1}{\color{MyPurple}0}{\color{SkyBlue}11}{\color{MyPurple}0}$.

Theorems & Definitions (39)

• Definition 1.1: DefantLiUngarian
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Example 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10