The Ungar Games
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916088831279104 |
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| author | Defant, Colin Kravitz, Noah Williams, Nathan |
| author_facet | Defant, Colin Kravitz, Noah Williams, Nathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06552 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Ungar Games Defant, Colin Kravitz, Noah Williams, Nathan Combinatorics 91A05, 91A46, 05A05, 05A15, 05A16, 05A17, 06B05, 06A07 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. |
| title | The Ungar Games |
| topic | Combinatorics 91A05, 91A46, 05A05, 05A15, 05A16, 05A17, 06B05, 06A07 |
| url | https://arxiv.org/abs/2302.06552 |